Physics: Principle of Least Action [3 ed.] 9781737915874, 1737915871

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Physics

Principle of Principle of Least Action Least Action Robert P. Massé Subtitle Kay. E. Massé Third Edition

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Also by Robert P. Massé

Copyright © 2022 Robert P. Massé ! Kay E. Massé

Physics: Nature of Physical Fields and Forces All rights reserved. Physics: Where It Went Wrong Vectors and Tensors of Physical Fields

Third Edition

Number Theory ISBN: 978-1-7379158-7-4 Complex Variables Laplace Transforms Massé, Robert P. Massé, Kay E. Physics: Principle of Least Action

i

Dedication This book is dedicated to the libraries of Florida

ii

Preface Mathematics is an essential tool in the continuing struggle to understand the physical world.

iii

!

The subject of this book is the principle of least action in

dynamics. This principle has a long history in physics and has now evolved into very powerful analysis techniques that are successfully employed in many branches of physics. Yet the basis for this principle has remained a mystery which continues

!

First Edition – October 26, 2021

!

Second Edition – February 22, 2022

!

Third Edition – March 8, 2022

to give rise to speculations of all sorts. In this book we will consider the basis of the principle of least action. !

We will begin with a review of the history of the principle of

least action. We will define action in physics and we will discuss metaphysical causes that have been proposed to explain the principle. We will present the mathematical implementation of the principle of least action as the calculus of variations. We will then derive Newton’s equations of motion from physical field theory and show how this leads to the principle of least action. !

Most of the sources for this book are given in references

at the end of the book. The research that forms the foundation of this book could not have been accomplished without the outstanding interlibrary loan programs of Florida.

!

Robert P. Massé!

Kay E. Massé

iv

!

3.4!

Poisson’s Field Equation of Gravity

!

3.5!

Aether Field Equation of Gravity

!

3.6!

Continuity of Aether Density in a Vacuum

!

3.7!

Force Density of Flowing Aether

1! Principle of Least Action

!

3.8!

Nature of Mechanical Force

!

1.1!

Least Action

!

3.9!

Nature of Gravitational Force

!

1.2!

Action Defined

!

3.10! Summary

!

1.3!

Hamilton’s Principle

!

1.4!

Cause of Least Action

4!

Newtonian Dynamics

!

1.5!

Summary

!

4.1!

Classical Concepts of Mass

!

4.2!

Nature of Mass

2! Variational Principles

!

4.3!

Mass Types

!

2.1!

Stationary Points and Paths

!

4.4!

Derivation of Newton’s Laws of Motion

!

2.2!

Functionals

!

4.5!

Derivation of Conservation Laws

!

2.3!

Calculus of Variations

!

4.6!

Nature of Inertia

!

2.4!

Constraints

!

4.7!

Absolute Motions and Galilean Relativity

!

2.5!

Classic Variational Problems

!

4.8!

Newtonian Dynamics

!

2.6!

Stationary Action

!

4.9!

Summary

!

2.7!

Summary

5!

Lagrangian Dynamics

3!

Nature of Physical Fields

!

5.1!

Euler-Lagrange Equation

!

3.1!

Physical Fields

!

5.2!

Degrees of Freedom

!

3.2!

Gravity

!

5.3!

Summary

!

3.3!

Newton’s Force Law of Gravity

Contents

v

6!

Foundations of the Principle of Least Action

!

6.1!

Physical Meaning of Action

!

6.2!

Basis of the Principle of Least Action

!

6.3!

Summary

Appendix A!

Vector Field Operations

!

A.1! Source/Sink Points and Field Points

!

A.2! Dirac Delta Function Appendix B!

The Greek Alphabet

Appendix C!

Vector Identities

Appendix D!

Integration by Parts

References

vi

Chapter 1 Principle of Least Action

S=

∫

mv ds

7

!

Since ancient times people have noticed that nature acts in

!

In this chapter we will review the history of the principle

the simplest ways. Nature is very efficient. Actions occurring

of least action. We will define action in physics and we will

within nature are such that minimum effort is required to

present Hamilton’s principle. Finally, we will discuss some of

achieve these actions. Much of the symmetry and ascetic beauty

the metaphysical theories that have been advanced in the past

we observe in the world about us has been attributed to this

for the basis of the principle of least action.

economy of effort by nature. The minimization of effort found in nature is known as the principle of least action.

1.1! LEAST ACTION

!

!

Variational techniques have been developed which have

Among the people who made notable contributions to the

proven to be very useful in solving many problems in physics.

development of the proposition that has become known as the

These techniques are all implementations of the principle of

principle of least action are: Hero of Alexandria, Dante, Fermat,

least action. In none of these techniques, however, is the basis

Newton, Euler, and Maupertuis.

for the principle of least action in dynamics revealed.

1.1.1!

!

In our quest to understand the universe, the physical

cause of the principle of least action has been sought. Despite centuries of searching, however, the physical cause of this principle has not been found. Nature and our Universe have only been personified as frugal, parsimonious, indolent, and lazy (see Bunn, 1995; Hildebrandt and Tromba, 1996; and Coopersmith, 2017). These are simply descriptive terms, and not causes. !

When no physical cause is discernible for natural events,

then metaphysical causes are often invoked. This has certainly been the case for the cause of least action in dynamics.

!

HERO

Hero of Alexandria (c. AD 10-70) is generally considered

to be the first person to propose that light travels over the shortest path (which implies least travel time). In his book Catoptrics he argued that: “Omnia enim quecumque ferunter continua uelocitate, hec in recta linea feruntur, sicut uidemus sagittas emissas

ab

arcubus.

Propter

uiolentiam

enim

emittentem conatur quod fertur ferri linea breuissima in distantia, non habens tempus tarditatis, ut et feratur linea maiori in distantia, non sinente uiolentia transmittente.

Propter

quod

utique,

propter 8

uelocitatem, conatur breuissima ferri. Recta autem est

”Le principe de Physique est que la nature fait ses

minima linearum habentium eadem ultima.”

mouvements par les voies les plus simple.”

“Whatever moves with unchanging speed, this moves in a

“The principle of physics is that nature makes its motions

straight line, as we see arrows sent from bows. For because

by the simplest way.”

of the impelling force it is carried over the shortest path, not having time for slowness, so as to be carried over a greater distance, since the impelling force does not allow it. And so, because of its speed, it moves over the shortest path. Now the shortest path having the same endpoints is the straight line.” (see Cohen and Drabkin, 1948; and Jones, 2001).

1.1.2! !

DANTE

Dante Alighieri (c. 1313) in a treatise supporting the

temporal equality of the pope and emperor stated:

In 1662 Fermat wrote a letter to Marin Cureau de la

Chambre in which he noted that light travels slower in an optically dense medium and that the propagation of light occurs in the least time. He defined his principle for light: “qui est qu’il n’y ait rien de si probable ni de si apparent que cette supposition, que la nature agit toujours par les moyens les plus aisés, c’est-à-dire ou par les lignes les plus courtes, lorqu’elles n’emportent pas plus de temps, ou en tout cas par le temps le plus court.” “which is that there is nothing so probable nor so apparent

”et omne superfluum Deo et nature displiceat, et omne

than this supposition, that nature acts always by the

quod Deo et nature displicet sit malum, ut manifestum

easiest means, that is to say by the shortest paths, when

est de se,”

they do not take more time, or in any case by the shortest

“and all superfluity displeases God and nature, and all that displeases God and nature is evil, as is self-evident”

1.1.3! !

!

FERMAT

time.” Fermat’s proposal became known as Fermat’s principle or the principle of least time.

Fermat thought that nature acts with simplicity. In 1657 he

stated: 9

“Since all effects of Nature follow a certain maximum or

1.1.4! !

NEWTON

minimum law, there is no doubt that, on curved paths,

As his first rule of reasoning in philosophy, Newton (1697)

which the bodies describe under the action of certain forces,

gave:

they possess some maximum or minimum property.” “We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances. To this purpose the philosophers say that Nature does nothing in vain, and more is in vain when less will serve; for Nature is pleased with simplicity, and affects not the pomp of superfluous causes.”

1.1.5! !

certain forces “ought to be a minimum” when the speed of the body was a function of position only.

1.1.6! !

MAUPERTUIS

Maupertuis (1740) showed that for a system of bodies at

rest, any disturbance of the equilibrium will result in the least

EULER

In 1744 Euler published a book titled A method for finding

curved lines enjoying properties of maximum or minimum, or the solution of isoperimetric problems in the broadest meaning. In the second appendix to this textbook he included a discussion of a principle of least motion. He stated: “Quoniam

He thought that the motion acquired by a body as a result of

omnes

naturæ

effectus

sequuntur

quandam maximi minimive legem; dubium est

possible change in action. Then in 1744 Maupertuis found the refraction of light it to be consistent with the general principle: “la Nature, dans la production de ses effets, agit toujours par les moyens les plus simples” “Nature, in the production of its results, acts always by the simplest methods” He noted that for light:

nullum, quin in lineis curvis, quas corpora projecta, si

“le chemin qu’elle tient est celui par lequel la quantité

a viribus quibuscunque sollicitentur, describunt,

d’action est la moindre.”

quæpiam

“the path that she takes is the one by which the quantity of

habeat.”

maximi

minimive

proprietas

locum

action is the least.” 10

He then explained what he meant by quantity of action:

“I have discovered the universal principle, upon which all

“Lorsqu’un corps est porté d’un point à un autre, il faut

the laws are founded; which extends equally to rigid bodies

pour cela une certaine action: cette action dépend de la

and elastic bodies; on which depend the motion and the rest

vîtesse qu’a le corps, & de l’espace qu’il parcourt”

of all material substances. It is the principle of the least

“When a body is carried from one point to another, it

quantity of action.”

requires a certain action: this action depends on the speed of the body and the distance that it travels”

He defined this general principle as: “Lors qu’il arrive quelque changement dans la Nature, la

and

Quantité d’Action, nécessaire pour causer quelque

“La quantité d’action est d’autant plus grande que la

changement dans la Nature, est le plus petite qu’il est

vîtesse du corps est plus grande, & que le chemin qu’il

possible”

parcourt est plus long; elle est proportionnelle à la somme

“When there occurs some change in Nature, the Quantity

des espaces multipliés chacun par la vîtesse avec laquelle le

of Action necessary to cause this change in Nature is the

corps les parcourt.”

least so it is possible.”

“The quantity of action is greater according as the speed of the body is greater, and the path that it travels is longer: it

He had now expanded his principle of least quantity of action

is proportional to the sum of the distances each multiplied

to cover not just light, but all motions. His principle came to be

by the speed with which the body travels through them.”

called the principle of least action.

In 1746 Maupertuis wrote: “j’ai découvert le principe universel, sur lequel toutes ces loix sont fondées; qui s’étend egalement aux çorps durs & aux corps élastiques; d’où dépend le mouvement & le repos

1.2! ACTION DEFINED !

We will now consider definitions of action as proposed by

Euler, Maupertius, and Hamilton.

de toutes les substances corporelles. C’est le principe de la moindre quantité d’action.” 11

1.2.1! !

minimum. He then proposed that the action S of a physical

ACTION OF EULER

Euler (1744) defined the quantity of motion S of a body

as:

body be defined as: !

S=m

!

∫ v ds !

(1.2-1)

S=

∫ L dt !

(1.2-3)

where L is the Lagrangian defined as:

L = T −V !

where m is the constant mass of a material body and v is its

!

speed.

where T is the kinetic energy and V is the potential energy of

(1.2-4)

the body. The motion of a body is then determined by finding

1.2.2! !

ACTION OF MAUPERTUIS

the stationary value of S as given by equation (1.2-3). Instead of

In 1746 Maupertuis defined the quantity of action S as:

S=

!

∫ m v ds !

(1.2-2)

where m is the mass of a material body and v is its speed. This definition of quantity of action and the definition of quantity of

the name principle of least action, he recommended that the proposition be called the law of stationary action (see Gelfand and Fomin, 1963).

1.3! HAMILTON’S PRINCIPLE

motion made earlier by Euler are essentially identical. Euler

!

(1753a) supported Maupertuis by arguing that the principle of

Hamilton’s principle:

least action of Maupertuis was a generalization of his own principle of least motion.

1.2.3!

ACTION OF HAMILTON

Hamilton’s law of stationary action is now known as

Of all the paths along which a body can possibly move in a given time interval, the path along which the body will move is determined by the stationary value of its action.

Considering reflected light, Hamilton (1833) recognized

That is, the motion of a body is entirely determined by the

that the action defined by Maupertuis may not always be a

stationary value of its action. To determine the stationary value

!

12

of the action, we can use a mathematical procedure known as

!

the calculus of variations (see Chapter 2).

universe, and such a universe must have the simplest and most

1.4! CAUSE OF LEAST ACTION !

Ascetic cause argues that a diety created a perfect

uniform form possible, which then is mirrored in all natural actions. This argument reflects human concepts of beauty and

We will now review metaphysical causes that have been

order. It is essentially the same argument advanced by Newton

proposed in the past for the principle of least action. These

in his first rule of reasoning in philosophy (see Section 1.1.4). It

causes are of two kinds: ascetic and theological. Both of these

follows from this cause that nature is economical, exerting only

types of causes have in common the concept of final cause.

a minimum effort to achieve any given effect.

1.4.1!

!

!

FINAL CAUSE

Final cause is a concept that was advanced by Aristotle to

explain certain naturally occurring events. The final cause concept states that the end result determines the action. Final cause is then a form of teleology. The principal of least action asserts that nature functions in a way that minimizes the effort required. This suggests that nature somehow has advance knowledge of the results of all possible efforts, and so chooses the action that will require the least effort. Final cause therefore

Theological cause assumes that some Higher Being has

designed Nature such that all its actions are minimized. Least action then exists simply due to the will of some diety. !

Arguments for a divinely created perfect universe date

back to antiquity. Plato in his Timaeus argues that God made the world spherical because this is the most perfect and most uniform structure possible, and uniformity is much better than non-uniformity. Similar arguments have been advanced by many people over the years.

attributes nature with knowledge.

1.4.2.1!

1.4.2!

!

METAPHYSICAL CAUSES OF LEAST ACTION

EULER

In 1744 Euler asserted:

Ascetic and theological causes of least action appeal to the

“Cum enim Mundi universi fabrica sit perfectissima,

existence of an all powerful Creator of the Universe. These are

atque a Creatore sapientissimo absoluta, nihil

metaphysical causes based on religious arguments.

omnino in mundo contingit, in quo non maximi

!

13

minimive

quæpiam

eluceat:

quamobrem

dubium prorsus est nullum, quin omnes Mundi

1.5! SUMMARY

effectus ex causis finalibus”

!

“Since the world’s entire structure is made the most

Universe has been known for many centuries, the physical

perfect by the wisest absolute Creator, nothing at all

cause of this property has remained a deep mystery. Given our

happens in the world for which no maximum or minimum

need to understand the nature of our Universe, it is not

rule is somehow evident: there is absolutely no doubt then

surprising that metaphysical causes have been resorted to as

that all actions in the world are from final causes”

explanations of the efficiency of nature when no other

1.4.2.2! !

ratio

MAUPERTUIS

While the existence of a minimizing property of the

explanation has been apparent.

Maupertuis (1746) described the origin of the principle

that he had discovered as follows: “C’est le principe de la moindre quantité d’action: principe si sage, si digne de l’Etre suprême, & auquel la Nature paroît si constamment attachée” “It is the principle of the least quantity of action: a principle so wise, so worthy of the supreme Being, and to which Nature appears so constantly tied” Maupertuis (1751) even concluded that since nature appears to follow the principle of least action which must be due to a wise creator, then this is proof of the existence of a God.

14

Chapter 2 Variational Principles

∂f d ∂f − =0 ∂y dx ∂ y′

15

!

In this chapter we will present the calculus of variations.

We will first review the concept of stationary points for a function and explain the concept of stationary paths. We will then define mathematical functionals. After presenting the calculus of variations, we will provide examples of this calculus using three classic variational problems.

2.1! STATIONARY POINTS AND PATHS !

Since we will encounter stationary paths in the calculus of

variations, we will now briefly review the analogous concept of stationary points of a smooth curve in differential calculus. We will consider a curve that can be described by a differentiable function y = y ( x ) of a single real variable x .

2.1.1! !

STATIONARY POINTS

A stationary point on a smooth curve is a point where the

Figure 2.1-1! Extremum points of a curve.

curve is neither increasing nor decreasing; it is stationary. The

!

Extremum points of a function can be local (within a given

curve must then have either a maximum, a minimum, or an

range of the function) or global (over the entire domain of the

inflection at a stationary point, and so the slope of the curve

function) as shown in the example in Figure 2.1-1. A maximum

will be zero at a stationary point (see Figure 2.1-1). Therefore a

or minimum of a function y = y ( x ) determined on the basis of

stationary point on a curve can be defined as any point where

derivatives of the function for a point x = x 0 can be either local

the derivative of the function representing the curve is zero.

or global since such extrema represent only changes in the

Maximum and minimum points are known as extremum

function y = y ( x ) that are in the neighborhood of x = x 0 .

points or extrema. 16

!

We have the following necessary and sufficient conditions

for an extremum to exist at a point x = x 0 : !

maximum exists if y′ ( x 0 ) = 0 and y′′ ( x 0 ) < 0

!

minimum exists if y′ ( x 0 ) = 0 and y′′ ( x 0 ) > 0

this is not possible since the right-sided and left-sided limits of a differentiable function must be the same. We can conclude that we must have y′ ( x 0 ) = 0 when x = x 0 is a maximum point.

We have the following necessary and sufficient conditions for an inflection to exist at a point x = x 0 : inflection exists if y′ ( x 0 ) = 0 , y′′ ( x 0 ) = 0 and y′′′ ( x 0 ) ≠ 0

!

Therefore the sign of y′ ( x 0 ) will depend on the sign of Δx . But

!

If the point x = x 0 is a minimum, a similar argument can

be made. !

■

Note that the symbol

■

signifies the end of a proof in this

book.

Proposition 2.1-1, Extremum Conditions for a Curve: Let y = y ( x ) be a real-valued differentiable function on an

interval [ x1, x 2 ] . If y = y ( x ) has an extremum at a point x = x 0 within the interval [ x1, x 2 ] , then y′ ( x 0 ) = 0 .

!

y ( x 0 + Δx ) − y ( x 0 )

Δx→ 0

Δx

!

(2.1-1)

stationary. The stationary path completely determines the stationary value of the integral.

2.2! FUNCTIONALS !

where Δx = x − x 0 . !

If the integral of a real-valued function over some path

calculated for all other possible paths, the path is called

The derivative of y = y ( x ) at point x = x 0 is given by:

y′ ( x 0 ) = lim

!

STATIONARY PATHS

results in a stationary value relative to the integral values

Proof: !

2.1.2!

If the point x = x 0 is a maximum for y = y ( x ) , then the

If a real-valued function is dependent not just on certain

independent variables, but also on a function or functions of

numerator of equation (2.1-1) will always be negative for

independent variables, then the real-valued function is called a

Δx ≠ 0 since we have:

functional. A functional is therefore a mapping from a set of

!

y ( x0 ) > y ( x0 + Δx ) !

(2.1-2)

functions into real numbers. Since an integral can map a 17

and so I is a function of y′ = y′ ( x ) which is a function of x .

function to a single real number, functionals generally take the form of integrals.

We then have I = I ( y′ ( x )) , and so I is a functional. If we wish

!

to determine the minimum length of a continuous curve

The concept and term of a functional arose from the

calculus of variations. Functionals are often encountered in

between the two points ( x1, y ( x1 )) and ( x2 , y ( x2 )) , we will

physics and engineering problems that involve curves or

then be minimizing a functional.

surface areas.

2.3! CALCULUS OF VARIATIONS

Example 2.2-1 Show that the length of a continuous curve y = y ( x ) between two given endpoints in the x-y plane is a functional.

!

A general method for determining extrema of certain

functionals (if such extremas exist) has been developed and is known as the calculus of variations. These functionals take the form of an integral I where:

Solution: We will let the endpoints be ( x1, y ( x1 )) and ( x2 , y ( x2 )) , and

I=

!

we will let I be the length of the curve. If ds is a differential

∫

x2

x1

f (Y ( x ) , Y ′ ( x ) , x ) dx !

(2.3-1)

element of length along the curve, we then calculate I by

This integral is not simply a function of discrete variables, but

integrating a connected set of straight line differential

is a function of the functions Y ( x ) and Y ′ ( x ) , making I a

elements ds : !

I=

∫

x2 x1

ds =

functional.

∫

x2

dx + dy =

x1

2

2

∫

x2 x1

2

⎛ dx ⎞ 1+ ⎜ ⎟ dx ⎝ dy ⎠

where we have used the Pythagoras theorem. We then have: !

I=

∫

x2 x1

1+ y′ 2 dx

2.3.1! ! set

PATHS OF INTEGRATION Each Y ( x ) represents a smooth path in the x-y plane. The of functions Y ( x ) is called a function space, and is the

domain of the functional I . All continuous paths connecting

the endpoints ( x1, y ( x1 )) and ( x2 , y ( x2 )) in the x-y plane are in 18

this domain. For some problems the endpoints may not be

where ε is a parameter having a very small absolute value, and

fixed, but for our purposes in this book they will be fixed.

where η( x ) is an arbitrary function that has a continuous

!

second-order derivative and is subject to the constraints:

The function

f

in equation (2.3-1) is taken to be

continuous and at least twice differentiable. For each of the paths Y ( x ) , integration of the function f yields a single real number I . The calculus of variations provides a method for finding from a set of paths Y ( x ) the single path y ( x ) for which

!

η( x1 ) = 0 !

η( x2 ) = 0 !

(2.3-3)

as shown in the example in Figure 2.3-1. Each ε represents a different path in the neighborhood of y ( x ) , with ε = 0 being the

the integral I will be a stationary value. Problems consisting of

path that makes the integral I stationary. By construction we

finding the stationary path y ( x ) from a set of possible paths

have:

Y ( x ) are characteristic of all variational problems.

!

!

Y ( x1 ) = y ( x1 ) = y1 !

Y ( x2 ) = y ( x2 ) = y2 !

(2.3-4)

Unlike the problem of determining a stationary point of a

known curve y ( x ) , we see that variational problems involve determining an entire path y ( x ) . Since each path is represented by an integral which is a real number, it will be necessary to find the stationary point of the curve composed of the set of real numbers obtained from all the functionals I . The stationary path y ( x ) will then be the path that makes the integral I a stationary point on this curve. !

We will now consider those smooth paths Y ( x ) that are

very close to the path y ( x ) of I , and that connect the endpoints

( x1, y ( x1 )) and ( x2 , y ( x2 )) . These neighboring paths to y ( x ) can

be represented by the equation: !

Y ( x ) = y ( x ) + ε η( x ) !

(2.3-2)

Figure 2.3-1! Stationary path y ( x ) (in red) and one of many neighboring paths Y ( x ) (in blue). 19

!

2.3.2! !

EULER EQUATION

when ε = 0 . The first-order derivative of I must then equal zero when ε = 0 :

Using equation (2.3-2), equation (2.3-1) becomes:

∫

I (ε) =

!

x2

x1

Since Y ( x ) = y ( x ) when ε = 0 , the integral I is stationary

f ( y ( x ) + ε η( x ) , y′ ( x ) + ε η′ ( x ) , x ) dx !

(2.3-5)

where I is a function only of ε after the integration has been

∂I ∂ε

!

f = f (Y , Y ′, x ) = f ( y + ε η, y′ + ε η′, x ) !

(2.3-6)

Taking the derivative of the integral I with respect to ε , we can differentiate under the integral sign since the integration limits do not depend on ε : ! !

∂I d = ∂ε ∂ε

∫

x2

f dx =

x1

∫

x2

∂f dx = ∂ε

∫

x2

x1

⎡ ∂ f ∂Y ∂ f ∂Y ′ ⎤ ⎢⎣ ∂Y ∂ε + ∂Y ′ ∂ε ⎥⎦ dx

!

∂I ∂ε

!

(2.3-7)

!

∫

x2

x1

ε=0

∫

x1

∂f ⎤ ⎡∂ f η + η′ ⎥ dx = 0 ! ⎢⎣ ∂Y ∂Y ′ ⎦

(2.3-11)

the second term in the integral by parts (see Appendix D) since have:

∂Y ′ = η′ ( x ) ! ∂ε

∂f ⎤ ⎡∂ f η + η′ dx ! ⎢⎣ ∂Y ∂Y ′ ⎥⎦

∂f ! ∂Y ′

!

u=

!

du =

(2.3-8)

d ∂f dx ! dx ∂Y ′

dv = η′ dx !

(2.3-12)

v = η!

(2.3-13)

we obtain:

and so:

∂I = ∂ε

=

x2

It is possible to remove η′ from this equation. We can integrate

From equation (2.3-2) we have: !

ε=0

the function f has continuous second-order derivatives. We x1

∂Y = η( x ) ! ∂ε

(2.3-10)

or

performed. We have: !

=0!

! (2.3-9) !

∂I ∂ε

=

∫

x2

x1

ε=0

!

⎡∂f ⎢⎣ ∂Y

x

2 ⎤ ⎡ ∂f ⎤ η⎥ dx + ⎢ η − ⎦ ⎣ ∂Y ′ ⎥⎦ x1

∫

x2

x1

⎡ d ∂f ⎤ ⎢⎣ dx ∂Y ′ η⎥⎦ dx = 0 (2.3-14) 20

Using equation (2.3-3) we then have: !

∂I ∂ε

= ε=0

∫

x2

x1

d ∂f ⎤ ⎡ ∂f − ⎢⎣ ∂Y dx ∂Y ′ ⎥⎦ η dx = 0 !

!

the calculus of variations. He devised a variational operator (2.3-15)

Because Y ( x ) = y ( x ) when ε = 0 , equation (2.3-15) can now be written as: !

∂I ∂ε

which acts to form the derivative of a functional, as distinguished from the ordinary derivative which acts to form the derivative of a function (see Section 2.3-4). Using the variational operator Lagrange was able to derive the Euler

= ε=0

∫

x2

x1

⎡ ∂f d ∂f ⎤ − ⎢ ∂y dx ∂ y′ ⎥ η dx = 0 ! ⎣ ⎦

(2.3-16)

Since η( x ) is an arbitrary function subject only to the endpoint contraints given in equation (2.3-3) and the requirement of differentiability, we have (see Proposition 2.3-1): !

Lagrange (1759, 1762, 1766) developed a new approach to

∂f d ∂f − = 0! ∂y dx ∂ y′

equation and generalize Euler’s methods for determining extrema of functionals. In recognition of Lagrange’s work, Euler designated this field of mathematics the calculus of variations. The Euler equation is now also known as the EulerLagrange equation. !

(2.3-17)

The Euler-Lagrange equation is not a sufficient condition

for I to be an extremum (an extremum may not even exist). For problems arising from dynamics, however, the existence of

This is the basic equation of the calculus of variations. This

extrema is generally evident. As Gelfand and Fomin (1963)

equation is a necessary condition that must be satisfied for the

note, “the existence of an extremum is often clear from the physical or

integral I to be stationary. A solution of this equation is called

geometric meaning of the problem”. Between any two possible

an extremal of the functional I .

states of a physical system there is always at least one shortest

!

path. Therefore we will not pursue mathematical proofs of

Equation (2.3-17) is called the Euler equation. It was first

derived by Euler using geometrical methods. In 1744 Euler

sufficiency conditions of extrema existence.

published a book on variational problems that became the

!

foundation of the calculus of variations. !

The Euler-Lagrange equation can be expanded as:

∂f ∂2 f ∂2 f ∂2 f − y′ − y′′ − =0! 2 ∂y ∂y ∂ y′ ∂x ∂ y ′ ∂ y′

(2.3-18) 21

!

From this equation we see that the Euler-Lagrange

equation is a second-order differential equation. Solution of this differential equation will include two arbitrary constants which

neighborhood α1 < xP < α 2 of xP within [ x1, x 2 ] for which f ( x )

has a constant sign since f ( x ) is continuous. Given that η( x ) is an arbitrary function, we can select η( x ) so that η( x ) > 0 on the

must be determined using the given endpoints of the path. The

interval [ α1, α 2 ] and η( x ) = 0 outside this interval. Such a

function defined by this equation is continuous on the interval

function is:

[ x1, x2 ] since

f is at least twice differentiable on this interval.

4 4 ⎧ ⎪ ( x − α1 ) ( x − α 2 ) η( x ) = ⎨ 0 ⎪⎩

!

2.3.3!

FUNDAMENTAL LEMMA OF THE CALCULUS OF VARIATIONS

Proposition 2.3-1, Fundamental Lemma of Calculus of Variations: If a function f ( x ) is continuous on an interval [ x1, x 2 ] and if:

∫

!

x2

f ( x ) η ( x ) dx = 0 !

(2.3-19)

x1

where η( x ) is an arbitrary differentiable function subject only to the constraints

η( x1 ) = 0 !

!

η( x2 ) = 0 !

(2.3-20)

α1 < x < α 2 x otherwise

We then have:

∫

!

x2

f ( x ) η ( x ) dx > 0 !

(2.3-21)

x1

This is a contradiction to the given equation (2.3-19) and so our assumption that at some point xP within the interval [ x1, x 2 ] we have f ( xP ) ≠ 0 must be false.

2.3.4! !

■

VARIATIONAL OPERATOR

From equation (2.3-8) we have:

∂Y = η( x ) ! ∂ε

then f ( x ) ≡ 0 on the interval [ x1, x 2 ] .

!

Proof:

Using this relation we can write equation (2.3-15) as:

!

We will assume contrarily that at some point xP within the

interval [ x1, x 2 ] we have f ( xP ) > 0 . There must then exist a

!

∂I dε = ∂ε

∫

x2

x1

(2.3-22)

d ∂ f ⎤ ∂Y ⎡ ∂f − ⎢⎣ ∂Y dx ∂Y ′ ⎥⎦ ∂ε dε dx !

(2.3-23) 22

We will now define the variation of I to be: !

∂I δI = dε ! ∂ε

for functionals of the differential operator d dx for functions. (2.3-24)

where δ is known as the δ operator or the variational operator. The δ operator was introduced by Lagrange when he developed an analytic approach to variational problems. Equation (2.3-24) is known as the first variation of the integral

I. !

We similarly have:

!

∂Y δY = dε ! ∂ε

(2.3-25)

From equation (2.3-2) we can write: !

Y ( x ) = y ( x ) + ε η( x ) = y ( x ) + δy ( x ) !

(2.3-27)

!

Y ′ ( x ) = y′ ( x ) + ε η′ ( x ) = y′ ( x ) + δ y′ ( x ) !

(2.3-28)

We then have: !

δy ( x ) = ε η( x ) !

(2.3-29)

!

δ y′ ( x ) = ε η′ ( x ) !

(2.3-30)

From equation (2.3-30) we have:

δ

!

and so equation (2.3-23) can be written: !

δI =

∫

x2

x1

d ∂f ⎤ ⎡ ∂f − ⎢⎣ ∂Y dx ∂Y ′ ⎥⎦ δY dx !

(2.3-26)

The integral I is stationary when its variation δI = 0 since the

dy ( x ) dx

=ε

d η( x ) ! dx

(2.3-31)

From equations (2.3-31) and (2.3-29) we can conclude:

δ

!

dy ( x ) dx

=

d δy ( x ) ! dx

(2.3-32)

slope of the curve I is then horizontal. Because the variation

and so the δ operator and the differential operator d dx

δY is arbitrary, the integrand must be zero when δI = 0 , and so

commute.

we again have the Euler-Lagrange equation.

!

We have not used the δ operator in the derivation of the

The δ operator represents a very small change from one

Euler-Lagrange equation in Section 2.3.2. The ε parameter

curve to a neighboring curve, while the differential operator d

makes it possible to obtain the Euler-Lagrange equation using

represents an infinitesimal change from one point to another

the ordinary partial derivative rather than the variational

point on the same curve. The δ operator is then the equivalent

operator.

!

23

2.3.5! !

We now multiply the Euler-Lagrange equation (2.3-17) by y′

NO EXPLICIT Y DEPENDENCE

If y does not explicitly appear in the function f , then from

y′

!

equation (2.3-17) we have:

d ∂f = 0! dx ∂ y′

!

!

(2.3-33) !

(2.3-34)

where C is a constant. Equation (2.3-34) is a differential equation of the first order. It provides an extremal of the Euler-

!

ANOTHER FORM OF THE EULER EQUATION

From equation (2.3-6) we have:

!

f = f ( y + ε η, y′ + ε η′, x ) !

(2.3-39)

d ⎛ ∂f ⎞ d ∂f ∂f y = y + y′′ ! ′ ′ dx ⎜⎝ ∂ y′ ⎟⎠ dx ∂ y′ ∂ y′

(2.3-40)

We can rewrite equation (2.3-39) as: !

!

∂f df d ∂f ∂f ! = y′ + y′′ + dx dx ∂ y′ ∂ y′ ∂x

Using:

Lagrange equation.

2.3.6!

(2.3-38)

From equations (2.3-38) and (2.3-37) we obtain:

or

∂f = C! ∂ y′

∂f d ∂f − y′ =0! ∂y dx ∂ y′

∂f df d ⎛ ∂f ⎞ ∂f ∂f ! = ⎜ y′ − y + y + ′′ ′′ dx dx ⎝ ∂ y′ ⎟⎠ ∂ y′ ∂ y′ ∂x

(2.3-41)

d ⎛ ∂f ⎞ ∂f ! f − y′ = dx ⎜⎝ ∂ y′ ⎟⎠ ∂x

(2.3-42)

or (2.3-35)

!

Therefore we can write: !

d f ∂ f dy ∂ f d y′ ∂ f ! = + + dx ∂y dx ∂ y′ dx ∂x

(2.3-36)

!

NO EXPLICIT X DEPENDENCE

We will now examine the case where the function f does

not depend explicitly on the independent variable x .

or !

2.3.7!

∂f ∂f df ∂f ! = y′ + y′′ + dx ∂y ∂ y′ ∂x

(2.3-37)

!

We can then write equation (2.3-42) as:

24

d ⎛ ∂f ⎞ f − y′ = 0 ! dx ⎜⎝ ∂ y′ ⎟⎠

!

(2.3-43)

This is a first-order differential equation. Integrating, we have

Let I be the length of the curve. From Example 2.2-1 we have: !

∫

I=

the Beltrami identity:

f−

!

∂f y′ = C ! ∂ y′

(2.3-44)

where C is a constant. This equation is also called a first

x2

1+ y′ 2 dx

x1

Letting f ( y, y′, x ) = 1+ y′ 2 we obtain: !

integral of the Euler-Lagrange equation.

∂f =0! ∂y

∂ ∂ 1+ y′ 2 = 0 ∂x ∂ y′

and so the Euler-Lagrange equation becomes:

2.3.8! !

NO EXPLICIT Y’ DEPENDENCE

If the function f does not depend explicitly on the

independent variable y′ . then the Euler-Lagrange equation (2.3-17) becomes:

∂f = 0! ∂y

!

(2.3-45)

which provides an extremal of the Euler-Lagrange equation. Example 2.3-1 Determine the shortest distance between two given points

( x1, y1 )

and ( x2 , y2 ) on a continuous curve y = y ( x ) in the x-y

plane.

!

y′ ⎞ ∂f d ∂f d ⎛ − = ⎜ ⎟ =0 ∂y dx ∂ y′ dx ⎜⎝ 1+ y′ 2 ⎟⎠

and so: !

y′ 1+ y′

2

=C

where C is a constant. Solving for y′ : !

y′ =

C 1− C

2

=m

where m is a constant. We have finally: !

y = mx+b

Solution: 25

where b is a constant. Therefore as expected a straight line is the shortest distance of a continuous curve between two

2.4! CONSTRAINTS

points in the x-y plane.

!

A constraint is a limit or condition that the solution of a

variational problem must satisfy. The calculus of variations can be used to determine stationary values when contraints exist.

Example 2.3-2 Determine the extrema of the functional: !

I=

∫

x2 x1

(y

2

)

− y′ 2 dx

We have f ( y, y′, x ) = y − y′ and so: 2

∂ ∂ 2 y′ = − 2 y′′ ∂x ∂ y′

and the Euler-Lagrange equation becomes: !

form:

!

∫

⎡⎣f (Y ( x ) , Y ′ ( x ) , x ) + λ φ (Y ( x ) , Y ′ ( x ) , x ) ⎤⎦ dx ! (2.4-2)

where Lagrange multipliers λ are used. Letting: !

y′′ + y = 0

!

y = C1 cos x + C2 sin x

I=

x2

x1

∂f d ∂f − = 2 y + 2 y′′ = 0 ∂y dx ∂ y′

and so the extrema are: !

(2.4-1)

second order derivatives. Equation (2.3-1) can then be written:

F = f (Y ( x ) , Y ′ ( x ) , x ) + λ φ (Y ( x ) , Y ′ ( x ) , x ) !

(2.4-3)

∂F d ∂F − = 0! ∂y dx ∂ y′

(2.4-4)

we have:

or !

φ (Y ( x ) , Y ′ ( x ) , x ) = 0 !

where φ is a continuous function with continuous first and 2

∂f = 2y! ∂y

We will consider the case where the constraint has the

!

Solution:

!

!

Equations (2.4-4) and (2.4-1) are then solved together. By incorporating the constraints into the function F , the solution for the constrained system can be found directly. 26

2.5! CLASSIC VARIATIONAL PROBLEMS !

Variational problems involve finding the maximum or

!

An isoperimetric problem is the determination of a plane

figure enclosing the largest possible area having a specified length perimeter. Queen Dido’s problem is considered to be a

minimum of a function or functional. Problems dealing with

classic isoperimetric problem.

extrema of functionals originated with a problem Johann

!

Queen Dido’s isoperimetric problem can be stated as:

!

For all curvilinear arcs of length L bounded by and having endpoints on a line, find the arc that encloses the maximal area.

Bernoulli used to challenge the mathematical community in 1696. We will now present Bernoulli’s problem and two other classic variational problems. These three problems are: 1.! Isoperimetric problems. 2.! Minimal surface of revolution problems. 3.! The brachistochrone problem.

2.5.1! !

ISOPERIMETRIC PROBLEMS

The isoperimetric problem is also known as Queen Dido’s

problem for historical reasons dating back to ancient Greece. In the epic poem the Aeneid Virgil describes how Queen Dido fled the Phoenician city of Tyre after a power struggle with her brother. Together with a group of supporters, she sailed to North Africa. The local authorities granted her as much land as can be enclosed by a single bull’s hide. She cut the hide into very thin strips which she used to bound a hill bordered by the Mediterranean coast. Upon this hill she founded the city of Carthage.

Figure 2.5-1! !

Two arcs each of length L .

We wish to find an arc of a given length L such that the

area enclosed by the curve composed of the arc and a straight 27

line will be maximal. Two arcs of length L are shown in Figure 2.5-1. We will show that the curve having the arc in red encloses the maximum area while the curve having any other arc (represented by the arc in blue) does not. !

The area A of the closed curve is given by:

A=

!

∫

L=

−a

∫

(2.5-1)

a

−a

1+ y′ 2 dx !

!

y′ =

dy ! dx

(2.5-2)

! (2.5-3)

The length L of the arc is a constraint, and so we will use

F = y + λ 1+ y′ 2 !

∂F d ∂F − = 0! ∂y dx ∂ y′

we obtain:

+ c1 !

(2.5-7)

y′ =

± ( x − c1 ) λ − ( x − c1 ) 2

2

!

(2.5-8)

y = ∓ λ 2 − ( x − c1 ) + c2 !

(2.5-9)

( y − c2 )2 = λ 2 − ( x − c1 )2 !

(2.5-10)

2

or !

Therefore we have: (2.5-4)

From the Euler-Lagrange equation: !

1+ y′

2

Integrating again we obtain:

Lagrange multipliers to write: !

y′

or !

where !

x=λ

a

y dx !

(2.5-6)

Integrating: !

The length L is given by: !

!

d ⎛ y′ ⎞ 1− ⎜ λ ⎟ = 0! 2 ⎟ dx ⎜⎝ 1+ y′ ⎠

!

( x − c1 )2 + ( y − c2 )2 = λ 2 !

(2.5-11)

This is a circle of radius λ . Since the points ( a, 0 ) and ( − a, 0 ) (2.5-5)

must be on the perimeter of the circle on opposite sides, the center of the circle is ( c1, c2 ) = ( 0, 0 ) , and so we have λ = a : !

x 2 + y2 = a 2 !

(2.5-12) 28

The perimeter of the circle is 2 L and so the solution to Queen Dido’s problem will be an arc in the form of a semicircle having arc length L = π a .

2.5.2! !

MINIMAL SURFACE OF REVOLUTION PROBLEMS

This type of problem was addressed by Newton (1687)

when he determined the solid of revolution having the minimum resistance to movement in a fluid (see Gould, 1985). Minimal surface of revolution problems are also known as soap film problems since soap film between two rings is known to assume a shape having minimum area. !

Minimal surface of revolution problems entail a surface of

revolution generated by rotating a curve y ( x ) about the x-axis.

Figure 2.5-2!

Surface of revolution for the curve extending from point P1 ( x1, y1 ) to point P2 ( x 2 , y2 ) .

The two endpoints P1 ( x1, y1 ) and P2 ( x 2 , y2 ) of the curve are fixed relative to each other as shown in Figure 2.5-2. ! !

Minimal surface of revolution problems can be stated as: Determine the curve that results in the minimal surface area of the surface of revolution for the curve.

The area element dS for the surface of revolution generated by

revolution of the curve extending from point P1 ( x1, y1 ) to point

P2 ( x 2 , y2 ) is:

!

dS = 2 π y ( x ) ds = 2 π y 1+ y′ 2 dx !

! !

The total surface area is then:

S=

∫

x2

2 π y 1+ y′ 2 dx !

(2.5-14)

x1

We wish to find the curve that results in the minimal surface area S of the surface of revolution. We will define: !

f ( y, y′ ) = 2 π y 1+ y′ 2 !

(2.5-15)

(2.5-13) 29

Since f ( y, y′ ) does not depend explicitly on the independent variable x , we can use the Beltrami identity given in equation (2.3-44): !

f−

∂f y′ = C ! ∂ y′

(2.5-16)

y 1+ y′ − 2

y y′ 2 1+ y′ 2

= C1 !

(2.5-17)

y 1+ y′

2

= C1 !

!

1+ y′ 2

=

(2.5-18)

(2.5-19)

2.5.3!

C1 1 dx ! = = 2 2 y′ dy y − C1

THE BRACHISTOCHRONE PROBLEM

Johann Bernoulli (1696, 1697) published a paper in which

he challenged mathematicians to solve the problem (which he had already solved) of determining the path of quickest descent vertical line. He named this path the brachistochrone (from the

(2.5-20)

Greek brachistos meaning shortest and chronos meaning time). This problem is regarded as the variational problem which led to the development of the calculus of variations.

or !

revolution. The curve is called a catenary (from the Latin catena

under gravity between two fixed points not located on the same

Solving for dx dy : !

(2.5-23)

This curve results in the minimum surface area of the surface of

!

C12 !

⎛ x − C2 ⎞ ! y = C1 cosh ⎜ ⎟ ⎝ C1 ⎠

and

catenoid.

or

y2

(2.5-22)

meaning chain), and the surface of revolution is called a

where we have defined C1 = C 2 π . Therefore: !

y + C2 ! C1

x = C1 cosh −1

!

!

We then have: !

We have finally:

dy dx ! = 2 2 C1 y − C1

Brachistochrone is a term originating from the Greek words (2.5-21)

‘brachistos’ meaning shortest and ‘chronos’ meaning time. Jacob Bernoulli (1697), Newton (1697), Liebniz (1697), and de 30

l’Hôpital

(1697)

were

all

able

to

solve

Bernoulli’s

brachistochrone problem. ! !

The brachistochrone problem can be stated as: A bead slides down a frictionless non-vertical wire under gravity. What shape should this wire have to provide the shortest travel time for the bead?

The bead slides down a frictionless wire from a point P1 ( x1 , y1 ) to a point P2 ( x2 , y2 ) as shown in Figure 2.5-3.

!

The travel time of the bead along an element of distance

ds of the wire is: ds ! dt = ! v

(2.5-24)

where v is the speed along ds . The initial speed is taken to be zero. !

From conservation of energy we obtain:

m g y1 =

!

1 m v2 + m g y ! 2

(2.5-25)

where m is the mass of the bead and g is the acceleration of gravity. We then have:

v = 2 g ( y1 − y ) !

!

(2.5-26)

The total travel time T of the bead on the wire is: !

T=

∫

P2

P1

ds = v

∫

P2

P1

dx 2 + dy 2 2g ( y1 − y )

=

∫

x2 x1

1+ y′ 2

2g ( y1 − y )

dx ! (2.5-27)

Let w = y1 − y . We then have: ! Figure 2.5-3! Bead slides down a frictionless wire.

T=

∫

x2 x1

1+ w′ 2 dx ! 2g w

(2.5-28)

We wish to find the curve for which T is a minimum. The integrand of equation (2.5-28) does not depend explicitly on the 31

independent variable x , and so we can use the Beltrami

Let w = k − k z where dw = − k dz . We then have:

identity in equation (2.3-48) to write: 2 1+ w′ 2 ∂ 1+ w′ − w′ =C! ∂w′ 2g w 2g w

!

! (2.5-29)

or

1+ w′ − 2g w

w′

!

!

(

2

2g w 1+ w′

2

)

=C (2.5-30)

and so:

!

!

(

2g w 1+ w′

2

)

=C (2.5-31)

!

w′ 2 + 1 =

(

2k ! w

(2.5-32)

)

!

w′ =

2k −1 ! w

dx = k tan 2

θ θ θ θ sin cos dθ = 2 k sin 2 dθ ! 2 2 2 2

(2.5-37)

dx = k (1− cosθ ) dθ !

(2.5-38)

x = k (θ − sin θ ) + C2 !

(2.5-39)

We also have by definition: (2.5-33)

!

w = y1 − y = k − k z = k (1− cosθ ) !

(2.5-40)

When θ = 0 then y = y1 and so x = x1 . Therefore from equation

so that: !

(2.5-36)

We then obtain: !

where 2 k = 1 C 2 g . Solving for w′ : 2

1− cosθ sin θ dθ ! 1+ cosθ

and so: !

We can write:

dx = k

or !

1

(2.5-35)

Let z = cosθ so that dz = − sin θ dθ . Therefore: !

2

1− z dz ! 1+ z

dx = − k

w dx = dw ! 2k − w

(2.5-39) we have: (2.5-34)

!

x1 = C2 !

(2.5-41) 32

so that:

the Euler-Lagrange equation (2.3-17) becomes:

!

x − x1 = k (θ − sin θ ) !

(2.5-42)

!

y − y1 = − k (1− cosθ ) !

(2.5-43)

!

! !

∂L d ∂L − = 0! ∂xi dt ∂ x!i

i = 1, 2, 3 !

(2.6-3)

This equation determines the least action that is a property

These two equations are the parametric equations of a

of our physical Universe. For Hamilton’s principle to be valid,

cycloid, which can be described as the locus of a point on the

the Euler-Lagrange equation (2.6-3) must then provide a valid

rim of a wheel that rolls without slipping. The wire should then have the shape of a cycloid to provide the shortest travel time for the bead. The bead moves faster down the beginning of the wire which is steeper, making the descent quicker than it would for a straight line path between the two points (even though the total distance traveled by the bead is greater than the straight line path between the two points).

We can use the calculus of variations to determine the

stationary action from Hamilton’s definition of action given in equation (1.2-3): !

S=

∫ L dt !

!

To determine the cause of the principle of least action, we

need to determine why equation (2.6-3) applies to our Universe. To do this we need to understand the nature of the physical fields that exist in our Universe. These fields will be examined in the next two chapters.

2.7! SUMMARY

2.6! STATIONARY ACTION !

description of dynamics in our Universe.

!

The calculus of variations provides an analytic method to

determine stationary values of functionals. Many different types of variational problems, including the determination of least action, can be solved using the calculus of variations.

(2.6-1)

Letting: !

f ( Y ( x ) , Y ′ ( x ) , x ) = L ( xi ( t ) , x!i ( t ) , t ) !

(2.6-2) 33

Chapter 3 Nature of Physical Fields

! ! 1 ∂ ! ! ⎡⎣∇ • (ζ υ ) ⎤⎦ = ∇ • g ζ ∂t

34

!

In this chapter we will present the important concept of

!

Considering these lines of force, Maxwell (1861a) stated

physical fields. We will define field particles and material point

"we cannot help thinking that in every place where we find these lines

particles. From a study of gravitational force fields, we will

of force, some physical state or action must exist in sufficient energy

determine the nature of the physical field that exists throughout

to produce the actual phenomena." Maxwell proceeded to "examine

our Universe. Knowledge of this physical field is essential to

magnetic phenomena from a mechanical point of view" based upon

understanding the basis for the principle of least action.

an interpretation of Faraday's concept of a physical medium of contiguous particles (Maxwell, 1861a, b).

3.1!

PHYSICAL FIELDS

!

Contiguous particles are simply particles packed so

The concept of a physical field was first developed by

closely together that there is essentially no space between them.

Euler in the 1750s during his studies of the kinematics of fluids

Faraday envisaged contiguous particles as being next to each

(see Truesdell, 1954). It was James Clerk Maxwell, however,

other, but not touching each other. He did not consider the

who more generally introduced the field concept into physics in

contiguous particles of his physical medium to be matter. He

the 1860s. Maxwell derived his field concept from the magnetic

noted that: “matter is not essential to the physical lines of magnetic

lines of force that Michael Faraday had proposed to explain

force any more than to a ray of light or heat.” Faraday’s ideas were

magnetic force patterns in space (Faraday, 1844, 1847, 1855).

used by Maxwell to formulate the concept of a continuous

Faraday employed the term magnetic field to denote the

physical field in which all actions occur only between

distribution of magnetic forces in a region (see Gooding, 1980).

contiguous particles. We will designate these contiguous

!

particles as field particles.

!

A magnetic line of force is a line having the direction of the

magnetic force at each point along the line. The concentration

!

A disturbance resulting from a change in the motion of a

of such lines in a given region indicates the intensity of the

field particle can then propagate to a distance only by a

force in the region. Faraday surmised that these lines of force

succession of actions between contiguous particles. Action-at-a-

exert an influence upon each other by transferring their action

distance is thereby reduced to actions between particles that are

from particle to contiguous particle of a physical medium.

contiguous. Some time duration is required for all these actions

35

to occur, and so field disturbances can only propagate with a

Each point particle of the continuum always retains its

velocity that is finite.

individual identity as well as all the physical properties

!

We see then that the existence of a continuous physical

(including the kinematical and dynamical properties) of its

field in a region requires the existence of a real physical

corresponding field particle except for volume and extension (a

medium consisting of contiguous particles. We will designate

point particle has neither). Nevertheless the same density and

this real physical medium as the field medium. The particles of

pressure are assumed to exist in the continuum as in the field

the field medium must be small enough so that a large number

medium.

of these field particles are contained within a volume element

!

that is considered infinitesimal relative to the volume of the

capable of being abstracted to form a mathematical continuum,

entire region.

this is not what defines a physical field. Rather, a physical field

While a physical field requires contiguous field particles

These contiguous field particles can then be used to form a

is some specific attribute associated with each of the field

continuous mathematical entity known as a continuum. This

particles being abstracted to form the mathematical continuum.

continuum is obtained by abstracting the field particles of the

A physical field is not a physical medium then, but is some

field medium to point particles, thereby allowing the field

attribute of the contiguous field particles that form the real

medium to be treated as a mathematical continuum filling a

physical medium. A physical field can never have an existence

region of space with no gaps between the points. The result is a

independent of the field particles that constitute the field

one-to-one correspondence between point particles of the

medium. A physical field is more therefore than simply a

continuum and geometrical points assigned to the field

mathematical device used to solve a physical problem. Physical

particles in the region of space occupied by the field medium.

fields are always real.

The mathematical continuum of abstracted physical particles

!

provides justification for using the limiting process of the

that is the physical field, several different types of physical

calculus on physical problems involving fields, and for using

fields can be identified. When the field particle attribute is a

differential equations to describe changes in the fields.

scalar quantity so that a scalar quantity is associated with each

Depending upon properties of the field particle attribute

point of the mathematical continuum, these quantities together 36

form a scalar field. Similarly, when the field particle attribute is

always valid, the superposition of different vector fields

a vector quantity or a tensor quantity so that a vector quantity

representing the same type of physical entity is always possible.

or a tensor quantity is associated with each point of the mathematical continuum, these quantities form a vector field or

3.2!

GRAVITY

a tensor field, respectively (in this book we will use the term

The search for an understanding of the physical force

tensor to refer to tensors of order two or higher). The

known as gravity has occupied philosophers and scientists for

temperature in the Earth’s atmosphere is an example of a

many centuries. Early concepts of force and matter developed

physical scalar field, and the wind velocity in the atmosphere is

in antiquity by Aristotle and other philosophers have come

an example of a physical vector field. The real medium that is

down through the ages and influenced the work of scientists

abstracted to a mathematical continuum in both these cases is

such as Kepler, Galileo, Huygens, and Newton (see Jammer,

air, and the field particles are extremely small air parcels.

1957, 1961). Advancement in science is, after all, an incremental

!

process with new discoveries resting on the foundation of

A field that does not vary with time (is independent of

time) is termed a stationary field or a steady-state field. A field

previous work.

that does vary with time is termed non-stationary.

!

!

Since vector fields can, by definition, vary from point to

gravity. We will begin by examining the gravitational force law

point, the vectors associated with a vector field are known as

of Newton, and then proceed to consider some field theories of

point vectors. They are defined for and occupy only a single

gravity.

In the following sections we will determine the nature of

point. Point vectors and line vectors are not the same then since all nonzero line vectors occupy more than a single point and

3.3!

NEWTON’S FORCE LAW OF GRAVITY

can be slid over many points. Therefore the magnitude of a

Newton’s approach to the study of gravity was to use “the

point vector cannot be equated with the length of the vector,

phenomena of motions to investigate the forces of nature, and then

unlike the magnitude of a line vector. Finally, since the

from these forces to demonstrate other phenomena.” From various

addition of two or more point vectors at any given point is

observations

and

experiments,

Newton

succeeded

in

determining particular propositions that: “are inferred from the 37

phenomena and afterwards rendered general by induction.” This type

always result in the occurrence of identical phenomena

of methodology works well in physics.

(Poincaré, 1905). A law in physics then denotes neither a

!

Isaac Newton’s great Philosophiae Naturalis Principia

requirement nor a prohibition, but only an observed pattern. In

Mathematica (known as Principia) was first published in 1687.

other words, a law in physics does not govern nature; it only

In this work Newton derived mathematical expressions for

describes nature. Therefore a law in physics is perhaps better

gravitational forces based upon his studies of pendulums,

considered an effect rather than a cause.

falling bodies, the orbits of planets, and the orbits of their moons. He found, for example, “that the forces by which the

3.3.1!

primary planets are continually drawn off from rectilinear motions,

!

and retained in their proper orbits, tend to the Sun” and that these

bodies. Newton (1687) defined mass to be a measure of the

forces “are reciprocally as the squares of the distances of the places of

quantity of matter in a material body. Newton thus introduced

those planets from the Sun’s center.” He also found that: “there is a

the concept of mass into physics (Cohen, 2002). Note that

power of gravity tending to all bodies, proportional to the several

Newton did not state that mass is the same thing as matter, but

quantities of matter which they contain.”

only that mass is a measure of the quantity of matter.

!

!

Newton had discovered the force law of gravity.

MATHEMATICAL EXPRESSION

Bodies containing matter are, by definition, material

Matter is the only entity in our Universe that possesses

Moreover, he had discovered that this force law of gravity

gravity. Mass, being only a measure of the quantity of matter,

applies not only to objects on Earth, but to objects in the

cannot possess gravity.

heavens as well. For example, he found that the Earth’s

!

gravitational force keeps the moon orbiting the Earth rather

masses m and M can be expressed in the form:

than flying off into deep space. !

A law in physics is a relation between physical entities

!

Newton’s force law of gravity for two material bodies of

F = −G

mM ! r2

(3.3-1)

that has been abstracted from empirical observations of these

where F is the gravitational attractive force between the

entities. In formulating laws of physics, the assumption is made

bodies, r is the distance between the bodies, and G is the

that the existence of identical physical circumstances will

gravitational constant of proportionality. The constant G was 38

first measured implicitly by Cavendish (1798) using a torsion

mass m is in the opposite direction of rˆ and so is directed

balance to determine the density of the Earth by weighing the

towards the body of mass M .

world. He did not calculate a numerical value for G , however (see Clotfelter, 1987; and Falconer, 1999). The first estimate of a numerical value for G was made almost one hundred years later by Boys (1894a, b, c, and d) using empirical procedures developed by Cavendish. Since then many measurements of G have been made. The current estimate of G is: ! !

G = 6.67428 × 10 −8 cm 3 sec-2 gm -1 !

(3.3-2)

Considering the force F on a material body of mass m

resulting from the gravitational pull of a material body of mass

M , we can rewrite equation (1.3-1) as: !

M F = −G 2 ! m r

In vector form, this can be written as: ! F M ! = − G 2 rˆ ! m r

(3.3-3)

(3.3-4)

where rˆ is a unit line vector that has its coordinate origin at the center of the body of mass M and that is directed along the line from this body to the body of mass m (see Figure 3.3-1). The  minus sign indicates that the force F acting on the body of

 Figure 3.3-1! Gravitational force F on a material body of mass m due to a material body of mass M .

3.3.2! !

PHYSICAL INTERPRETATION

Newton’s force law of gravity given in equation (3.3-1) can

be interpreted physically to imply that some sort of action-at-adistance is occurring as each of the two material bodies exerts an attractive force on the other; that is, the two material bodies will tend to accelerate towards each other. Action-at-a-distance is defined here to mean that two bodies separated in space and 39

having no intervening medium exert a physical effect on each

noted that he used “the words attraction, impulse or propensity of

other across the empty space between them.

any sort towards a center, interchangeably, one for another;

!

Since time is not a parameter in equation (3.3-1), time does

considering those forces not physically, but mathematically.” He was

not enter into Newton’s force law of gravity. We then have the

not attempting “to define the kind or the manner of any action, the

additional implication that the action-at-a-distance of gravity

causes or the physical reason thereof,” nor was he attributing

occurs instantaneously. Therefore gravitational force appears to

“forces, in a true and physical sense, to certain centers (which are

propagate with infinite speed according to Newton’s force law

only mathematical points)” when he referred to ”centers as

of gravity.

attracting, or as endued with attractive powers.” Moreover, Newton

!

On the basis of our everyday experience, this physical

(1693b) wrote that the very thought that “one body may act upon

interpretation of gravity does not appear plausible. As Dolbear

another at a distance through a vacuum without the mediation of any

(1897) stated ". . . every body moves because it is pushed, and the

thing else by and through which their action or force may be conveyed

mechanical antecedent of every kind of phenomena is to be looked for

from one to another is to me so great an absurdity that I believe no

in some adjacent body possessing energy; that is, the ability to push or

man who has in philosophical matters any competent faculty of

produce pressure." It is hard to imagine how a force could act

thinking can ever fall into it.” Newton clearly did not believe in

over some large distance without an intervening medium. It is

action-at-a-distance. He felt the need for a medium that serves

even harder to believe that this action could be instantaneous.

to transmit force (Jourdain, 1915c). Therefore Newton’s theory

We must conclude, therefore, that Newton’s force law of

of gravity clearly does not postulate instantaneous action-at-a-

gravity expressed in the form of equation (3.3-1) does not lend

distance.

itself to a direct physical interpretation as to the cause of

!

gravity. This prompts us to ask what Newton thought about the

qualities of forces, but investigating the quantities and mathematical

relation he had discovered for gravitational forces.

proportions of them” in his Principia. Therefore we see that

!

Newton stated very clearly in his Principia that he

Newton considered his force law of gravity to be useful for

purposed “only to give a mathematical representation of those forces,

mathematically calculating gravitational forces, but not for

without considering their physical causes and seats.” Newton also

determining their physical causes.

Newton realized that he was not defining the “physical

40

according to the laws which we have explained, and abundantly

3.3.3! !

PHYSICAL CAUSE OF GRAVITY

serves to account for all the motions of the celestial bodies, and of our

Newton did, however, attempt to find the cause of gravity.

sea” (Newton, 1687). He concluded that the “cause of gravity is

He was certain that gravity “must proceed from a cause that

what I do not pretend to know, and therefore would take more time to

penetrates to the very centers of the Sun and planets, without

consider of it“ (Newton, 1693a).

suffering the least diminution of its force; that operates not according to the quantity of the surfaces of the particles upon which it acts (as

3.4!

POISSON’S FIELD EQUATION OF GRAVITY

mechanical causes do), but according to the quantity of solid matter which they contain, and that propagates in all directions to immense

!

distances, decreasing always by the square of the distances.” He

Principia was first published, Siméon Denis Poisson derived an

suspected that the ultimate cause of gravity was to be found in

equation that can be used to describe the gravitational field

“an ethereal medium”, a physical nonmaterial medium thought

resulting from a material body. Using vector analysis methods,

to permeate all space (Newton, 1730; and Jammer, 1957). He

we will now show how Poisson’s field equation of gravity can

thought that gravity was the result of this ethereal medium

be obtained from Newton’s force law of gravity. Subsequently

flowing with accelerated motion into material bodies and

we will consider the physical interpretation of this field

then collecting within these bodies (Aiton, 1969). Newton

equation.

In 1813, more than one hundred years after Newton’s

thought the ethereal medium to be composed of distinct particles and to be mechanical in nature (see Hall and Hall,

3.4.1!

GRAVITATIONAL FIELD

1960).

!

!

Nevertheless, Newton could not verify the cause of

masses M and m as shown in Figure 3.3-1. We will define a

gravity, and he was not prepared to hypothesize on its physical

material point particle to be a material body that has been

cause: “But hitherto I have not been able to discover the cause of those

abstracted to an infinitesimal size so that: “the distances between

properties of gravity from phenomena, and I frame no hypotheses; . . .

its different parts may be neglected” as Maxwell (1877) suggested.

And to us it is enough that gravity does really exist, and act

A material point particle can then be considered to have no

We will now once again consider two material bodies of

41

volume or extension, and so it will occupy a single point in

the material body of mass m be a material point particle.

of the body of mass M if the point particle is placed at a point  in space specified by the position vector r (see Figure 3.4-1).  Since g is a real physical vector quantity that can be measured,  we can conclude that the vector field g surrounding the body

Newton formulated his gravitational theory in terms of such

of mass M is a real physical vector field.

space. We will assume, nevertheless, that a material point particle has density and that this density is constant. We will let

material point particles. We will specify the location of this  particle in space by the line position vector r that has its coordinate system origin in the center of the body of mass M . !

In order to avoid the action-at-a-distance concept of force,

we will now consider that at all points in space surrounding the material body of mass M a gravitational acceleration field due to the matter in this body exists. This gravitational acceleration field of the material body of mass M exists in space whether or not any material point particle is present at a particular point in space to experience the acceleration.  The vector quantity F m in equation (3.3-4) has the dimensions of acceleration and represents the gravitational acceleration existing around the material body of mass M . We  will define F m to be the gravitational field intensity or   gravitational field g . A vector quantity g can be associated therefore with each point of space surrounding the material body of mass M .

 The vector g is the acceleration that a material point

particle of mass m will experience due to the gravitational pull

Figure 3.4-1!

 Line position vector r from the material body of mass M to a body of mass m .

!

Using Newton’s law of gravity as given in equation   (3.3-4), we can then determine F m (and so g ) for all points in space about the body of mass M : 42

!

! F ! M ≡ g = − G 2 rˆ ! m r

where the unit vector rˆ is given by: ! ! r r ! rˆ = ! = ! r r

(3.4-1)

3.4.2! !

MATHEMATICAL EXPRESSION

The density of a substance is defined as the quantity of

the substance per unit volume. If ρ is matter density and Qm (3.4-2)

 Equation (3.4-1) shows that the acceleration g of a body of mass m due to the gravitational pull of a material body of mass M is a function of M , but is independent of m . !

If the material body of mass M is stationary, its  gravitational field g will also be stationary. A material point particle of mass m placed in such a stationary field will  instantaneously experience the gravitational acceleration F m of the field. We see now that the reason gravitational force appears to propagate with infinite speed according to Newton’s

is quantity of matter, then matter density must be Qm per unit volume, and so for a material body: !

Qm =

∫∫∫ ρ dV !

(3.4-3)

V

where V is the volume of the material body. It is important to notice that matter density ρ is not defined in terms of mass. Since for a material body mass has been found to be a measure of the quantity of matter, we have: !

Qm =

∫∫∫ ρ dV ≈ M !

(3.4-4)

V

force law of gravity is that this law effectively assumes the

where M is the mass of the material body. In fact, mass “serves

gravitational field is stationary, and so gravitational force

for measuring a portion of matter so well that matter and mass appear

already exists at all points within the field. The gravitational

to be synonyms” as noted by Rougier (1921). However, mass and

force that is associated with a stationary gravitational field will

matter are not the same type of thing (see Section 4.2). For

always be constant (not a function of time) at any given point

example, electrons do not contain matter but still have mass.

within the field. It is then not correct to state that Newton’s law

Only for a free neutron does the mass of a material body

of gravity assumes that gravitational force propagates with an

exactly equal the quantity of matter of the body (see Massé,

infinite speed.

2022).

43

!

Instead of considering the entire material body of mass

M , we will now consider only a single very small volume element ΔV of this body. The matter density ρ of this volume element can be taken as constant since the dimensions of ΔV are taken to be very small. The mass ΔM of the matter contained within ΔV is then given by: ! !

ΔM =

∫∫∫

ρ dV = ρ

ΔV

∫∫∫

!

dV = ρ ΔV !

(3.4-5)

  To obtain the gravitational field g at a point r in space

(3.4-8)

where δ ( r ) is the Dirac delta function (see Appendix A), and so we can rewrite equation (3.4-7) as: !

ΔV

! rˆ ∇ • 2 = 4 π δ (r ) ! r

! ! ∇• g = − G ρ

∫∫∫ 4 π δ (r ) dV

!

(3.4-9)

ΔV

We then have finally:   ! ∇ • g = − 4π G ρ !

(3.4-10)

due to the gravitational pull of only the matter of mass ΔM

This equation, which is known as Gauss's law of gravity or as

contained within the volume element ΔV , we can use  equations (3.4-1) and (3.4-5) to express the gravitational field g

Gauss's flux theorem of gravity, describes the gravitational  field g due to matter of mass ΔM and density ρ contained

as:

within a very small volume element ΔV .

!

! g = −Gρ

∫∫∫

ΔV

rˆ ! 2 dV r

(3.4-6)

!

The law of gravitation defined by Newton’s force law of  gravity is linear. Therefore the gravitational field g due to each

The factor rˆ r 2 is under the integral sign since the coordinate  origin of the position vector r is taken to be in the volume  element ΔV . The divergence of the gravitational field g is then

volume element of a material body can be calculated

determined by the relation:

gravitational field of the entire material body. In other words,  the gravitational field at a point r due to a body of mass M can   be obtained by summation of g at the point r for all the

! ! ! ! ∇• g = − G ρ∇•

∫∫∫

ΔV

rˆ dV = − G ρ r2

! rˆ ∇ • 2 dV ! (3.4-7) r ΔV

∫∫∫

as is shown in Appendix A (the matter density ρ is constant). From vector analysis we have:

individually using Newton’s force law of gravity, and then a vector summation can be performed to determine the total

volume elements ΔV in the body of mass M . If this body is  located far from the point r , it is possible to use equation 44

(3.4-10) directly for the entire mass M with ρ representing the

which is Poisson’s field equation of gravity for matter of mass

average density of the body.

ΔM and density ρ contained within a volume element ΔV .

!

!

For a stationary gravitational field resulting from matter of

Poisson’s field equation describes the gravitational field

mass ΔM contained in a volume element ΔV , we can define

existing around a material body of matter density

the Newtonian stationary gravitational potential ϕ at a point  r by: ΔM ! ϕ ≡G ! (3.4-11) r

Mathematically we must obtain the same results from Poisson’s

We then have: ! ΔM ! ∇ϕ = − G 2 rˆ ! r

since Newton’s force law of gravity is not a function of time, (3.4-12)

ρ.

field equation of gravity (3.4-15) as from Newton’s force law of gravity (3.3-1) since, as just shown, Newton’s force law can be used to derive Poisson’s gravitational field equation. Moreover, Poisson’s field equation of gravity (3.4-15) only applies to gravitational fields that are stationary. Therefore Poisson’s field

and so, for matter of mass ΔM contained in a volume element

equation of gravity and Newton’s force law of gravity must be

ΔV , we can use equations (3.4-1) and (3.4-12) to write the  stationary gravitational field g in the form:

mathematically equivalent for a stationary gravitational field.

!

! ΔM ! g = − G 2 rˆ = ∇ϕ ! r

(3.4-13)

3.4.3! !

PHYSICAL INTERPRETATION

The physical interpretation of Poisson’s field equation of

We see therefore that it is possible to express a stationary  gravitational field g in terms of the gradient of a scalar

gravity (3.4-15) is, however, very different from the physical

potential function ϕ . From equations (3.4-10) and (3.4-13) we

than the action-at-a-distance interpretation of gravity obtained

have:

from Newton’s force law of gravity, Poisson’s field equation is

! or !

! ! ! ! ∇ • g = ∇ • ∇ϕ = − 4 π G ρ ! ! ! ∇ • g = ∇ 2ϕ = − 4 π G ρ !

(3.4-14)

interpretation of Newton’s force law of gravity (3.3-1). Rather

based upon and must be interpreted in terms of the continuous  gravitational field g that exists in space about any material body.

(3.4-15) 45

  Since g is a physical field, g must be an attribute of real  field particles of a real field medium. Since g is a vector

attribute of the field particles that is the gravitational field.

acceleration field, this attribute of the particles must be

we can conclude that the field medium, aether, is real.

!

Since the gravitational field is a real physical acceleration field,

acceleration. We can conclude then that, at each point of the gravitational field, real particles must be in accelerated motion.

3.4.4!

We can also conclude, therefore, that a real physical medium

!

must exist in space and that some flowing motion must be

associated with the field particles of a real field medium as was

occurring within this medium to produce the observed

noted in Section 3.1. If within the region of the vector field there

gravitational acceleration.

exists an entity that produces a discontinuity in the particular

!

This is analogous to observing the effects of wind velocity,

particle attribute that is the vector field, then such an entity is

which is a vector velocity field, and concluding that a real

known as a source or a sink for the field (see Kellogg, 1929).

physical medium (air) must exist and must be in motion. For

Only vector fields have field sources and sinks. From vector

example, when we observe through a closed window the

analysis, we know that a vector equation in the form of ! equation (3.4-10) can always be written for a vector field ϒ

autumn leaves blowing about, we know that air is in motion although we cannot directly see the air. !

The real physical medium whose flowing motion

produces the gravitational field must be consistent with Poisson’s field equation (3.4-15). We will refer to this real field medium as aether for historical reasons to be discussed in Section 3.4.6. Aether particles are then the field particles of gravitational fields. The mathematical continuum for the gravitational acceleration field is abstracted from aether particles with each aether particle corresponding to a point in

VECTOR FIELD SOURCES AND SINKS

A physical vector field consists of some vector attribute

having a source or a sink. We have then: ! ! ! ∇ • ϒ = ± 4π K ρ !

(3.4-16)

where the left side of this equation is the divergence of a vector ! field ϒ (vector attribute of field particles), and where the right side of this equation has a positive sign for a source and a negative sign for a sink. The parameter K is a constant required to make the units consistent. The density ρ is a scalar that is the strength of the source or sink per unit volume, and so is a

the gravitational field. The acceleration of aether particles is the 46

measure of the effect the source or the sink has on the vector ! field ϒ .

designate a field non-flow source to be a field positive source

!

For the special case where the attribute that constitutes the

(recognizing that non-flow sources and sinks can nullify each

vector field is a kinematic flow property (flow velocity or flow

other). We can summarize by noting that there exist two types

acceleration) of the field medium, sources and sinks of the

of vector field sources and, correspondingly, two types of

vector field can be interpreted as places where this real physical

vector field sinks:

and a field non-flow sink to be a field negative source

field medium is being continuously created and continuously

1.! Field flow sources and field flow sinks.

destroyed, respectively (Lamb, 1879; and Granger, 1985). For

2.! Field non-flow sources and field non-flow sinks (field positive sources and field negative sources).

such cases, we will designate any source as a field flow source and any sink as a field flow sink. When the rate at which a source continuously creates a field medium (or a sink

3.4.5!

continuously destroys a field medium) is constant, the source

MATTER AS A FIELD FLOW SINK FOR AETHER

(or sink) is referred to as steady.

!

!

If the vector field does not consist of a kinematic flow

density of a material body is ρ > 0 and since the sign is

attribute of the field medium, then the source and sink will not

negative on the right side of equation (3.4-15), ρ must represent

represent places where the field medium is being, respectively,

the strength of a sink for the gravitational field. Since ρ is the

created and destroyed. That is, they will not represent a field

density of matter within the volume element ΔV of a material

flow source or a field flow sink. Rather, both source and sink

body, the sink for the gravitational field is then matter. The field

will produce the attribute constituting the vector field. The only

medium for a gravitational field is aether, and the attribute of

constraint is: when a source and a sink of this type and of equal

aether

strength come together, they can nullify each other. We will

Gravitational acceleration is a kinematic flow property of

designate this type of source as a field non-flow source and the

aether: flow acceleration. Therefore matter is a field flow sink

corresponding sink as a field non-flow sink. Since a field non-

for aether. Poisson’s field equation (3.4-15) for the gravitational

From equations (3.4-15) and (3.4-16), we see that since the

that

is

the

gravitational

field

is

acceleration.

flow sink is really just another kind of source, we will also 47

field is then an equation describing the accelerating flow of

upon other particles of aether except in the very few places

aether into matter.

where ordinary matter exists. He conceived of aether as a

!

While matter is a field flow sink for a gravitational field,

medium that can move very rapidly towards Earth. He thought

there is no known entity that is a field flow source for a

that aether is capable of transmitting force so that all action-at-

gravitational field. Other fields such as electromagnetic fields

a-distance can be explained as the result of forces acting

do have both source and sink entities.

between contiguous aether particles. He also considered that

3.4.6! !

AETHER

aether provided the means by which light is propagated. Following Descartes, Huygens (1690a, b) postulated aether as

The concept of a real physical medium called aether (or

the nonmaterial physical medium in which light waves

ether) is not new. The origins of aether as a metaphysical

propagate. Newton (1693b) thought that: “It is inconceivable, that

concept can be traced back to antiquity. The word ‘aether’ is

inanimate brute matter should, without the mediation of something

from a Greek word meaning ‘the upper purer air’. Aether was

else, which is not material, operate upon and affect other matter

considered by Aristotle (c. 384 BCE to 322 BCE) to be the

without mutual contact, . . .” He considered aether to be the cause

“primary” substance of the Universe, and the substance that fills

of gravity and of certain wavelike properties evident in the

up all empty parts of space. He made aether his quintessence

reflection and refraction of light.

(fifth essence) of the Universe, completing the essences of air,

!

water, fire and earth.

physical nonmaterial medium called aether. The basis for this

!

The scientific concept of aether has existed for almost four

support was often one of the following: to avoid the action-at-a-

hundred years, beginning in the seventeenth century with

distance explanation of physical forces (such as gravitational,

Kepler (1620) and Descartes (1637, 1638, 1644). Kepler thought

electrical, and magnetic forces), or to provide a real medium for

that the space between planets is filled with aether, whereas

light waves to propagate in. For example, Euler (1746, 1768,

Descartes thought that aether permeates the entire Universe,

1772) thought that all space was filled with ether and that: “rays

filling all voids where matter does not exist. Descartes believed

of light are nothing else but the shakings or vibrations transmitted by

Since then many scientists have supported the concept of a

that particles of aether form a continuum, everywhere pressing 48

the ether.” These are the historical reasons that we are calling the physical medium introduced in the previous section ‘aether’.

3.4.6.1!

!

!

After the existence of aether as a physical medium was

AETHER IS AN INVISCID FLUID

Using our interpretation of Poisson’s field equation of

first proposed, numerous aether theories were developed to

gravity as describing the accelerating flow of aether into matter,

explain various physical phenomena. Nevertheless the physical

we are now able to determine some of the physical properties

properties and the functions of aether remained unclear and,

of aether. For aether to be the physical medium whose motion

for most of the twentieth century, the very existence of aether

results in the observed gravitational field, aether must be able

was doubted (see Massé, 2022).

to flow rapidly. Aether is therefore a fluid.

!

!

Before considering the nature of aether, we need to

The flow of aether associated with gravity appears to be

provide a definition of a fluid. A fluid can be defined as a

ideal since there is no evidence of internal friction in aether.

physical medium that can flow (a fluid cannot sustain shear

Observations of spectral lines of stellar light exhibit no increase

stress). A solid can be defined as a physical medium that

in blurring with distance to the star. Light waves can propagate

maintains a definite shape and volume due to large cohesive

across the vast distances of space without any attenuation

internal forces. A solid is the frozen state of a fluid. All

except that due to geometrical spreading. Therefore aether

material media can flow at some temperature, and so all

must be perfectly elastic. No internal cohesion exists within

material media have a fluid state. If a fluid has no internal

aether. As Sir Oliver Lodge (1923) noted “ether has nothing of the

friction, it is considered to be capable of ideal flow.

nature of viscosity.” He also stated (1925a) “Ether fritters away no

!

energy, it preserves all: it is perfectly transparent; it transmits light

Viscosity of a fluid is a measure of its internal resistance

to gradual deformation by shear stress (resistance to flow). This internal resistance to gradual deformation is due to

from the most distant stars without waste or loss of any kind.”

internal cohesion. A fluid having no viscosity is known as an

3.4.6.2!

AETHER IS UBIQUITOUS AND CONTINUOUS

inviscid fluid, and the flow of such a fluid is known as inviscid

!

flow.

so the field medium of gravity (aether) must exist wherever

A gravitational field is always associated with matter, and

there is matter. Moreover, aether must be continuous down to 49

some very small dimension since the gravitational field appears

ultimate particles of all homogeneous bodies are perfectly alike.” If

to be continuous. We can conclude therefore that aether must

aether were not homogeneous, we would expect variations to

be ubiquitous and continuous in our Universe. Only at

be observed in both protons and neutrons since they are field

dimensions approaching the size of the aether particles is the

flow sinks for aether.

aether discontinuous. We will designate an aether particle as an aetheron. !

As J. J. Thomson (1909) noted, matter occupies “but an

insignificant fraction of the universe, it forms but minute islands in the great ocean of the ether, the substance with which the whole universe is filled.” Lodge (1925a) also noted: “The first thing to realise about the ether is its absolute continuity.” While some scientists around the beginning of the twentieth century thought that aether was not discontinuous at any dimension (see Doran, 1975), aether actually consists of extremely small physical particles.

3.4.6.3!

AETHER IS HOMOGENEOUS

3.4.6.4! !

NUCLEONS ARE STEADY SINKS FOR AETHER

Since nucleons do not vary with time, they must constitute

steady sinks for the field medium, aether. Therefore aether must flow continuously and uniformly into matter.

3.4.6.5! !

THE FLUID AETHER HAS HIGH DENSITY

Given the continuity of aether and the very small size of

aetherons, we can conclude that the density of aether must be extremely high. In comparison, the density of any material body is very low. The distances between the nuclei of atoms in a material body are very great relative to the dimensions of aetherons. Moreover, wherever there are nucleons, the aether

Since matter, which is a field flow sink for aether, exists in

flows into these nucleons. This explains how aether can be very

the form of nucleons (protons and neutrons), aether must

dense and yet flow through material bodies without any

appear continuous at the dimensions of nucleons. This requires

apparent resistance. We recall Newton's words for the cause of

particles of aether to have dimensions that are orders of

gravity: it “penetrates to the very centers of the Sun and planets,

magnitude smaller than nucleons. Moreover, since all protons

without suffering the least diminution of its force“ (see Section

are identical and all neutrons are identical, aether must also be

3.3.3).

!

homogeneous in composition. As Dalton (1808) noted, “the 50

3.4.6.6! !

AETHER IS NEARLY INCOMPRESSIBLE

Since aether consists of extremely small contiguous

3.4.6.8! !

AETHER HAS RIGIDITY

While aether has the flow properties of a perfectly

particles and so is an extremely dense medium, aether is nearly

frictionless fluid, it also has the elastic properties of a perfectly

incompressible. The compressibility of a physical substance is

elastic solid. This is evident since light waves, which are

a measure of the relative change in volume that occurs in the

transverse elastic waves, propagate in aether. Transverse waves

substance as a result of an applied force. The physical

can only propagate in a medium that has rigidity (which is

properties of aether causing it to be nearly incompressible

defined as resistance to relative motion between particles of

generally lead to the flow of aether being incompressible. In

the medium). Therefore aether must have rigidity.

fluid dynamics, incompressibility describes fluid flow in which

!

no change in fluid volume occurs. When the flow of a fluid is

electromagnetism binding atoms together (while their nuclei

incompressible, the fluid’s density remains constant.

remain widely separated by a vacuum). This binding force

The rigidity of material solid bodies is due to the force of

generally prevents material solid bodies from flowing (it

3.4.6.7! !

AETHER IS NONMATERIAL

We can also conclude that aether must be entirely different

provides the mechanism whereby material bodies can exist in the frozen state known as solid).

from matter since matter is a field flow sink for aether. The

!

For aether to possess a frozen state would require that a

existence of aether must then be independent of matter. We see

binding force exist between aether particles. Since aether is

therefore that aether is not a material medium; aether is

perfectly elastic and flows easily, no binding forces can exist

nonmaterial. Nevertheless, aether is a real physical medium.

between aether particles. Without such binding forces aether

Because aether is nonmaterial, it has no material particles that

cannot have a frozen state, and so aether does not exist as a

can be in motion, and so it cannot possess temperature; aether

solid. Therefore the rigidity of aether is not due to aether being

is perfectly cold.

in a frozen state. Rather, the rigidity of aether must be due to its extremely high density whereby all aether particles are contiguous with other aether particles. Any given aether 51

particle has almost no room to move relative to other aether

aether is frictionless or ideal. When there exists a void into

particles since very little void space exists between aether

which aether particles can flow without any relative motion

particles.

by

occurring between contiguous aether particles, aether will flow

electromagnetic forces, but by neighboring aetherons that are

freely (uniformly) from a region of high pressure to a region of

contiguous.

low pressure. Because the flow of aether is always inviscid, if it

!

We see therefore that the mere possession of rigidity by a

is also incompressible, then aether will act as a perfect fluid. No

medium is not a sufficient criterion to indicate whether the

deformation will occur, and so no stresses resulting from elastic

medium is a fluid or a solid. The physical cause of the rigidity

restoring forces will be present.

Aetherons

are

held

in

place

then,

not

must also be considered. When this is done we see that, although aether has rigidity (a property previously thought to

3.4.7!

PHYSICAL CAUSE OF GRAVITY

characterize only solids), aether is a fluid and not a solid.

!

Material fluids do not possess rigidity, but the nonmaterial

flowing into matter (see Sections 3.4.3 and 3.4.5). Matter is

fluid, aether, does. This explains why it is that when relative

required for the generation of a gravitational field. Since

displacements of contiguous aether particles occur, the fluid

matter is a gravitational field flow sink, matter is a sink into

aether acts as a perfectly elastic solid. The rigidity of aether

which the gravitational field medium (aether) is actually

causes elastic restoring forces to result when any displacement

flowing. For a spherical material body, the direction of aether

of the aether from an equilibrium position occurs.

flow into the body is vertical across the entire surface of the

Gravitational acceleration is the acceleration of aether

body. Aether does not flow out of matter because matter is not a

3.4.6.9! !

AETHER FLOW IS FRICTIONLESS

Given the cause of aether rigidity, it is clear that no

field flow source for aether. Aether, therefore, must be consumed at a steady rate by any nucleon.

internal resistance exists for the flowing motion of aether (since

!

there is then no relative motion between contiguous aetherons).

incompressible, the density of aether can be expected to remain

Therefore aether has zero viscosity and so is inviscid.

constant during the flow of aether into matter, except in

Moreover, aether has no internal friction, and so the flow of

proximity to nucleons where the density of aether will increase.

Since aether has extremely high density and is nearly

52

The flow of aether will then be incompressible when not in

surrounding the body of mass M

proximity to nucleons, and nearly incompressible when in

constantly disappearing into the body.

proximity to nucleons. As the flow of aether converges into

!

matter, the near incompressibility of aether causes an increase

as shown in Figure 3.3-1, the aether flow into both the body of

in the flow velocity of aether so that the aether density can

mass M and the body of mass m will draw towards the bodies

remain constant and the flow can remain incompressible (until

the aether that exists between them; as a result a negative

in proximity to a nucleon). It is just this increasing flow rate of

pressure gradient will be created in the aether. Since no aether

aether as it approaches matter that creates the observed

flows out of the material bodies, the flow of aether into the two

gravitational acceleration field (see the discussion in Section

bodies will cause the bodies to be pushed closer together.

3.6.4 on the continuity of aether flow).

Moreover, the converging flow of aether into each body will

We can conclude then that the gravitational acceleration field of a material body is caused by three factors:

!

1.!

Aether being ubiquitous in our Universe.

2.!

Matter being a field flow sink for aether.

3.!

Aether being nearly incompressible.

If we once again consider the material body of mass M

shown in Figure 3.3-1, we can conclude that aether must flow

to replace the aether

If a second material body of mass m is within this region

result in an increase in velocity of the aether, and so also of the two material bodies. In other words, the acceleration of aether towards the material bodies results in the acceleration of the two material bodies towards each other. This is gravitational attraction. The force of gravity, therefore, is the action of contiguous particles of aether, and is not some action-at-adistance. Lodge (1908) was correct when he stated: “Matter acts on matter only through the ether.”

continuously into the body from all directions. This aether flow

We also see that two material bodies do not directly

will draw towards the body of mass M the aether that exists in

attract each other; the apparent attraction is due to the flow of

the surrounding region. A negative pressure gradient in the

aether into matter creating a pressure differential. As Lodge

aether will result (see Section 3.7), and this negative pressure

(1908) noted, “when the mechanism of attraction is understood, it

gradient will cause a continuous flow of aether into the region

will be found that a body really only moves because it is pushed by something from behind.” Without the physical medium aether, 53

gravitational force would not exist. Flowing aether carries with

the flow of aether explains the results of Galileo Galilei’s

it any material body within the flow. This answers the age-old

famous experiments with free falling bodies from which he

question of how matter can appear to act where it is not. Matter

determined that the gravitational acceleration of a body is

acts through the aether it disturbs.

independent of the body’s mass and composition. From his

!

From Poisson’s field equation (3.4-15), we see that the flow

experiments, Galileo came to the conclusion “that in a medium

of aether into a material body of mass M is proportional to the

totally devoid of resistance all bodies would fall with the same

matter density ρ of the body. Matter density is the field flow

speed” (Galilei, 1638).

sink strength density for the field medium aether. Obviously

!

the greater the matter density of a body, the greater will be the acceleration in the flow of aether into the body.

bodies is independent of their mass, we see from equation  (3.4-1) that gravitational force F will always be proportional to

!

Since mass is a measure of the quantity of matter in a

the mass m of the body on which it acts. Since matter is a field

material body, mass is a measure of field flow sink strength.

flow sink and not a field flow source for aether, the

The acceleration of aether flowing into a body of mass M can

gravitational acceleration field associated with a material body

then be expressed as a function of M . We also see that the acceleration of a body of mass m resulting from the body being

is always directed towards and never away from the body.  Therefore gravitational force F will always be attractive.

drawn along by aether flowing into a body of mass M will be

!

dependent on the mass M , but not on the mass m . The

visualized as a liquid flowing towards a drain. The motion of a

acceleration of the body of mass m is due entirely to (and is

material body of mass m towards the body of mass M is in

equal to) the acceleration of the aether flowing towards the

some ways similar to a cork floating in the liquid, and being

body of mass M .

drawn towards the drain. Obviously the motion of the cork will

!

Aether flow, therefore, very simply explains what is

be independent of the cork’s mass. The increase in fluid speed

perhaps the most fundamental property of gravity: why the

as it approaches the drain will cause the floating cork to

acceleration experienced by a body in a gravitational field is not

accelerate towards the drain. This analogy is not perfect since

a function of the quantity of matter in the body. In other words,

the body of mass M is entirely surrounded in all directions by

 Because the gravitational acceleration g experienced by all

The aether flow into a material body of mass M can be

54

aether flowing into it. Moreover, a material body of mass m

an inverse distance-squared law of attraction could be obtained

remains a field flow sink itself as it moves toward the field flow

by simply considering ‘negative’ matter to be a field flow sink

sink of mass M .

for aether, and matter to be a field flow source of aether (see

The aether that flows continually into matter is consumed

Kragh, 2002). Also Kirkwood (1953, 1954) noted that the

and does not flow out. Aether flowing into matter must then

acceleration field of gravity is “strongly suggestive of the flow of a

replace lost energy since matter is in equilibrium with the flow

fluid medium,” and he considered this fluid medium to be

of aether into it (we know from observation that the quantity of

aether.

matter in a material body does not increase nor decrease with

!

time if the body is left undisturbed).

have determined that matter is a field flow sink for aether and

!

continually expends energy. For matter to exist in the form of

that the acceleration of flowing aether particles constitutes the  gravitational acceleration field g . This provides a very simple

nucleons, a continuous and steady flow of aether into matter

physical cause for gravity. Since gravitational acceleration is

must be required. This can also be inferred from the fact that

real, the field medium, aether, and the field flow sink for aether,

matter can be defined as a field flow sink for aether. We see

matter, must both also be real. We see too that the laws of fluid

then that gravitation reveals some aspects of the essence of

dynamics as applied to a perfect fluid are appropriate for

matter. Without aether no material bodies could exist in our

describing the mechanism of gravitation.

Therefore we can conclude that matter by simply existing

Universe. !

A physical cause of gravity involving sinks or sources of

3.5!

In summary, from Poisson’s field equation of gravity we

aether has previously been suggested in a number of studies

AETHER FIELD EQUATION OF GRAVITY

[for example, Riemann, 1853; Thomson, 1870 (as discussed by

!

With the insight into the physical cause of gravity

Ball, 1892); Pearson, 1891; and Ellis, 1973, 1974]. Riemann was

obtained from Poisson’s field equation of gravity, we will now

aware that gravity could be explained by the continuous flow

develop a new gravitational field equation. This field equation,

of aether into every material particle, and that this requires

based upon the concept of flowing aether, provides more

material particles to be aether flow sinks. Pearson showed that 55

detailed information concerning the physical processes that

single fluid point particle of fixed identity in its trajectory, we

produce gravitational fields.

will be using a Lagrangian specification of the flow. !

3.5.1!

MATHEMATICAL EXPRESSION

We will define aether density ζ as the quantity of aether

per unit volume. Aether density ζ is then analogous to matter

fluid parcel that has been abstracted to an infinitesimal size so

density ρ (see Section 3.4.2). Therefore aether momentum per  unit volume or aether momentum density ζ υ is analogous to

that its dimensions can be neglected. A fluid particle can then

the momentum density of matter. The change with time in

be considered to have no volume or extension. Unlike a material point particle, however, the density of a fluid point

aether momentum density for a fluid point particle of aether as ! it flows along is ∂(ζ υ ) ∂t .

particle is not necessarily constant.

!

We will define a fluid point particle to be a very small

Since the gravitational force due to matter within ΔV is

We will now consider matter of mass ΔM and density ρ

proportional to ΔM , the time derivative of the momentum

contained within a very small spherical volume element ΔV .

density of aether flowing through the surface area ΔS of ΔV

We will take the matter in ΔV to be the reference frame (see  Section 4.7). Aether at a point r outside of ΔV is flowing

must also be proportional to ΔM . This is consistent with the

!

continuously towards the matter within ΔV with a flow   velocity υ that is a function of r , the position vector whose origin is centered in the matter within ΔV . We will focus on a very small parcel of the aether flowing towards the matter within ΔV . We will abstract this very small parcel of aether to an infinitesimal size, making it a fluid point particle. Note that a fluid point particle of aether is not an aetheron, but is an abstraction of many aetherons. Since in this section we will be describing fluid flow by tracing the dynamical history of a

following facts: 1.! Matter within ΔV is a field flow sink for aether. 2.! Nearly incompressible inviscid flow of aether towards this sink causes fluid point particles of aether to accelerate as they approach the sink. Their momentum will change with time in proportion to the strength of the sink. 3.! Mass ΔM is a measure of the quantity of matter (strength of the sink) within ΔV . We can therefore write: 56

!

−

∫∫

∂ ! (ζ υ ) • nˆ dS = 4 π G k ΔM ! ∂t

(3.5-1)

Taking the time derivative, we obtain:

spherical volume element ΔV which has surface area ΔS . The

! ⎡ ! ∂ζ ! ⎤ ∇ • ⎢ζ α + υ ⎥ = − 4 π G k ρ ! (3.5-6) ∂t ⎣ ⎦   where α is the time variation ∂υ ∂t of the aether flow velocity.

constant of proportionality in this equation is taken to be

Using the vector identity given in equation (C-16) of Appendix

4 π G k where G is the gravitational constant of proportionality

C, this equation can also be written as:

ΔS

!

where nˆ is the outward directed unit normal vector to the

from Newton’s force law of gravity given in equation (3.3-1), and k is an undetermined constant. Using equation (3.4-5), we can also write equation (3.5-1) as: !

−

∫∫

ΔS

!

! ! ! ! ! ⎡ ∂ζ ! ⎤ ζ ∇ • α + α • ∇ζ + ∇ • ⎢ υ ⎥ = − 4 π G k ρ ! ⎣ ∂t ⎦

(3.5-7)

!

∂ ! (ζ υ ) • nˆ dS = 4 π G k ∂t

∫∫∫

ΔV

ρ dV !

(3.5-2)

To determine the constant k , we will now assume that the  gravitational field is stationary. We will consider a point r that is not in proximity to nucleons and that is on the flow trajectory

From Gauss’s theorem given in equation (C-59) of Appendix C,

(pathline or particle path) of a given fluid point particle of

we can rewrite the left side of equation (3.5-2) as:

aether. As was noted in Section 3.4.7, the flow of aether in a

!

−

∫∫

ΔS

∂ ! (ζ υ ) • nˆ dS = − ∂t

! ∂ ! ∇ • (ζ υ ) dV ! ∂t ΔV

∫∫∫

gravitational field is incompressible when not in proximity to (3.5-3)

constant:

and so equation (3.5-2) becomes: !

! ∂ ! ∇ • (ζ υ ) dV = − 4 π G k ∂t ΔV

∫∫∫

∫∫∫

ΔV

ρ dV !

(3.5-4)

Since ΔV is arbitrary, we must have: !

 ∂  ∇ • (ζ υ ) = − 4 π G k ρ ! ∂t

nucleons. Therefore the density of aether in such a flow will be

(3.5-5)

!

∂ζ =0! ∂t

(3.5-8)

!

  ∇ζ = 0 !

(3.5-9)

Equation (3.5-7) can then be written as: ! ! ! ζ ∇ •α = − 4π G k ρ !

(3.5-10) 57

Using equation (3.4-10), equation (3.5-10) becomes: ! ! ! ! ! ζ ∇ •α = k ∇ • g ! !

! (3.5-11)

 At any point r not in proximity to nucleons, the aether

flow acceleration toward the matter of mass ΔM will simply  equal α because of the relations given in equations (3.5-8) and  (3.5-9). Since the gravitational acceleration g is just the aether flow acceleration as noted in Section 3.4.7, we must therefore have: !

  α = g!

(3.5-12)

1 ! ⎡ ! ∂ζ ! ⎤ ! ! ∇ • ⎢ζ α + υ ⎥ = ∇ • g ! ζ ∂t ⎦ ⎣

 Equations (3.5-16) and (3.5-17) describe the gravitational field g  at a point r completely in terms of properties of the aether at this point. Since equations (3.5-14) through (3.5-17) are simply different forms of the same equation, each of these equations can be considered to be the aether field equation of gravity. ! !

Equation (3.5-4) can now be written as:

∫∫∫

ΔV

and so the constant k is: !

k =ζ !

(3.5-13)

Equations (3.5-5) and (3.5-6) can then be written as: !

1 ! ∂ ! ∇ • (ζ υ ) = − 4 π G ρ ! ζ ∂t

(3.5-17)

∂ ! ! ⎡⎣∇ • (ζ υ ) ⎤⎦ dV = − 4 π G ζ ∂t

∫∫∫

ΔV

ρ dV !

(3.5-18)

where we have interchanged the time and space differential operations and have used the equality k = ζ . We note that the volume flux of an entity is defined to be a scalar that specifies

(3.5-14)

the quantity of the entity flowing per unit time per unit   volume. In equation (3.5-18), the term ∇ • (ζ υ ) is the aether flux (quantity of aether flowing per unit time per unit volume) into

!

1 ! ⎡ ! ∂ζ ! ⎤ ∇ • ⎢ζ α + υ ⎥ = − 4 π G ρ ! ζ ∂t ⎦ ⎣

(3.5-15)

Using equation (3.4-10), we can also write equations (3.5-14) and (3.5-15) as: !

! ! 1 ! ∂ ! ∇ • (ζ υ ) = ∇ • g ! ζ ∂t

(3.5-16)

ΔV . The aether field equations of gravity (3.5-14) and (3.5-16) can be written in terms of aether flux: !

1 ∂ ! ! ⎡⎣∇ • (ζ υ ) ⎤⎦ = − 4 π G ρ ! ζ ∂t

(3.5-19)

!

! ! 1 ∂ ! ! ⎡⎣∇ • (ζ υ ) ⎤⎦ = ∇ • g ! ζ ∂t

(3.5-20) 58

The units of G are cm 3 s-2 gm -1 as given in equation (3.3-2).

medium, aether, into matter. The same physical interpretation

These units can be seen to be correct and perhaps more

pertains to Poisson’s field equation of gravity. This agreement

intuitive from equation (3.5-19) than from equation (3.3-1).

in interpretation is to be expected since the aether field

!

The flux vector of an entity is defined to be a vector

equation of gravity is based upon findings obtained from

representing the momentum density of the entity (the entity

Poisson’s field equation of gravity.   ! The gravitational field g at a point r is completely

density multiplied by the entity velocity), and so is the quantity of the entity flowing per unit time through a unit ! area. The gravitational flux vector Λ G is then the quantity of

determined by properties of the aether at this point as can be

aether flowing per unit time through a unit area and is given

gravitational fields must then reside entirely within the aether.

by:

! ! ΛG = ζ υ !

!

(3.5-21)

1 ∂ ! ! ⎡∇ • Λ G ⎤⎦ = − 4 π G ρ ! ζ ∂t ⎣ ! ! 1 ∂ ! ! ⎡⎣∇ • Λ G ⎤⎦ = ∇ • g ! ζ ∂t

!

3.5.3! !

and equations (3.5-19) and (3.5-20) can be written as: !

seen from equations (3.5-16) and (3.5-20). The energy of

PHYSICAL CAUSE OF GRAVITY

Gravitational acceleration is caused by aether flow that is

accelerating towards matter. As we have seen, this accelerating (3.5-22)

flow of aether results from matter being a field flow sink for the nearly incompressible aether. We will designate all aether flow (incompressible and nearly incompressible) resulting from

(3.5-23)

aether flowing into matter as gravitational aether flow. !

Matter has an associated gravitational acceleration field

Bridgman (1941) stated that the need in physics for a

only because there is an accelerated flow of aether into matter

gravitational flux vector is imperative.

resulting from matter being a field flow sink for aether. The

3.5.2! !

PHYSICAL INTERPRETATION

field particles for gravitational fields are aetherons, and the flow acceleration of aetherons is the attribute of the field

The aether field equation of gravity can be interpreted

particles that is the gravitational field. Finally, we note that the

physically as describing the flow acceleration of a real physical

aether field equation of gravity is completely consistent with 59

both Newton’s force law of gravity and Poisson’s field equation

!

of gravity. Moreover the aether field equation of gravity shows

Q=

∫∫∫

ζ dV !

(3.6-1)

ΔV

us that the gravitational field is linear.

where ζ is the aether density. This equation can be compared

!

If aether flow is accelerating without the presence of

with equation (3.4-3). Because the volume element ΔV itself is

matter, an acceleration field will still exist. We will denote such

not a function of time, the rate of any increase of the total

an acceleration field as an aether acceleration field, and we will

quantity of aether Q within the volume element ΔV is given

reserve use of the term gravitational acceleration field for

by:

accelerations resulting from gravitational aether flow.

3.6! !

CONTINUITY OF AETHER DENSITY IN A VACUUM We will now consider the flow of aether in a region of

space having no field flow sinks for aether. Such a region of space contains a vacuum since matter is the only field flow sink for aether, and absence of matter defines a vacuum. We will describe the aether flow in the vacuum of this region of space by considering a very small volume element ΔV of vacuum. We will let the volume element ΔV itself remain constant with

!

∂Q = ∂t

∫∫∫

ΔV

∂ζ dV ! ∂t

(3.6-2)

Since we are considering a region of space that has no field flow sinks for aether, and since no field flow sources of aether are known to exist in our Universe, there will be no sinks or sources for aether flow within ΔV . Therefore ∂Q ∂t must equal the rate of influx of aether through the surface area ΔS bounding ΔV : !

∂Q =− ∂t

∫∫

ΔS

! ζ υ • nˆ dS = −

∫∫

ΔS

! ! ζ υ • dS !

(3.6-3)

element. Such a volume element is known as a control volume

where nˆ is the outward directed unit normal vector to the  surface area ΔS , and υ is the velocity of aether flowing into

in fluid mechanics (see Granger, 1985), and the open surface of

ΔV . Invoking Gauss’s theorem given in equation (C-59) of

the control volume is known as the control surface.

Appendix C, we can rewrite the relation for ∂Q ∂t given in

time, although aether can flow freely through the surface of the

The quantity of aether Q in the volume element ΔV of

equation (3.6-3) as:

vacuum is: 60

∂Q =− ∂t

!

∫∫∫

ΔV

! ! ∇ • (ζ υ ) dV !

(3.6-4)

From equations (3.6-2) and (3.6-4) we then have:

∫∫∫

!

ΔV

! ⎤ ⎡ ∂ζ ! + ∇ • ζ υ ( )⎥ dV = 0 ! ⎢⎣ ∂t ⎦

! (3.6-5)

Since the volume element ΔV is arbitrary within the vacuum, we obtain the equation of continuity for aether density:

∂ζ   + ∇ • (ζ υ ) = 0 ! ∂t

! !

a small constant volume element ΔV that is moving with a  velocity υ :

(3.6-6)

From this equation of continuity for aether density, we see

that the rate at which the density of aether ζ increases within a very small volume element ΔV will equal the convergence of

  ∂ζ   + υ • ∇ζ + ζ ∇ • υ = 0 ! ∂t

(3.6-7)

The first two terms of equation (3.6-7) can be defined to be: !

Dζ ∂ζ ! ! ≡ + υ • ∇ζ ! Dt ∂t

(3.6-8)

where Dζ Dt is the total rate of change or total derivative of aether density ζ within the constant volume element ΔV . ! !

The derivative D Dt is defined as: D ∂ ! ! ≡ +υ •∇ ! Dt ∂t

(3.6-9)

flux of aether into ΔV (assuming no sinks or sources for aether

When applied to some physical quantity, the derivative D Dt is

are present within ΔV ). In fact, this equation can be interpreted

known as the substantive derivative, substantial derivative,

as stating that no sinks or sources for aether exist within ΔV .

total derivative, or material derivative of the physical quantity.

Similar continuity equations apply for any physical density

This derivative is calculated with respect to a coordinate system

entity that has no sinks or sources within ΔV (and so the physical density entity cannot be destroyed or created within

attached to a constant volume element ΔV moving with  velocity υ . Because the derivative D Dt is a scalar operator, it

ΔV ).

can be applied to scalars, vectors, and tensors.

3.6.1! !

THE SUBSTANTIVE DERIVATIVE

Expanding equation (3.6-6) using the vector identity given

in equation (C-16) of Appendix C, we have for any point within

!

The substantive derivative consists of two operator terms:

the term ∂ ∂t is the temporal rate of change and is known as the local time derivative or the Eulerian derivative; the term   υ • ∇ is the spatial rate of change and is known as the 61

convective derivative. The local time derivative represents the

Dζ =0! Dt

!

change with respect to time of the physical quantity, and the

(3.6-11)

quantity with respect to spatial position within the flow. The

This equation can be considered to be the definition of  incompressible flow for a fluid of density ζ and flow velocity υ

spatial rate of change results from the convection of the

(Kellogg, 1929). For such a flow, we have from equations

constant volume element ΔV from one position within the

(3.6-11) and (3.6-10):   ! ∇ •υ = 0 !

convective derivative represents the change of the physical

medium (where the physical quantity has one value) to a different position within the medium (where the physical quantity may have another value). For a steady-state medium, the local time derivative will always be zero. If no convection is occurring or if the medium is homogeneous, the convective derivative will always be zero.

3.6.2! ! !

INCOMPRESSIBLE FLOW OF AETHER IN A VACUUM

From equations (3.6-7) and (3.6-8) we obtain:

  Dζ + ζ ∇ •υ = 0 ! Dt

(3.6-10)

When flowing aether is not in proximity to a nucleon, no compression of the aether will occur, and so aether density will be constant. From equations (3.5-8), (3.5-9), and (3.6-8), we obtain the equation for the incompressible flow of aether:

(3.6-12)

and so the incompressible flow of aether is solenoidal. Equation (3.6-12) will always be true for any fluid whose flow is incompressible. While aether is not completely incompressible, the flow of aether is incompressible except in proximity to nucleons. !

When the flow of aether is incompressible, the aether

density ζ remains constant by definition. Equation (3.6-2) can then be written as: !

∂Q = ∂t

∂ζ dV = 0 ! ∂t ΔV

∫∫∫

(3.6-13)

and so from equations (3.6-3) and (3.6-4), we have: !

∂Q = −ζ ∂t

∫∫

! ! υ • dS = − ζ

ΔS

∫∫∫

! ! ∇ • υ dV = 0 !

(3.6-14)

ΔV

where we once again obtain equation (3.6-12). Therefore the quantity of aether within any constant volume element ΔV (not 62

containing field flow sinks or sources) remains constant for the

constant at each point. Streamlines will then vary with time,

incompressible flow of aether.

and will not coincide with fluid point particle trajectories.

3.6.3!

3.6.4!

!

STREAMLINES OF AETHER FLOW IN A VACUUM

Aether flow is described in terms of a velocity field. We

!

CONTINUITY OF AETHER FLOW IN A VACUUM

A surface enclosing a portion of a fluid’s streamlines will

will now consider incompressible aether flow in a vacuum

have the shape of a tube and is known as a stream tube.

along a line in the flow known as a streamline in fluid

Maxwell (1855) noted that for stream tubes “Since this surface is

dynamics. If at any given instant in a flowing fluid, the tangent  to a line is in the direction of the velocity vector υ of the fluid at

generated by lines in the direction of fluid motion no part of the fluid

each of the points of the line, then the line is defined as a

the fluid as a real tube.” The boundary of a stream tube consists

streamline of the flow. Streamlines will then always indicate

of streamlines.

instantaneous fluid flow direction. By definition, therefore, no

!

flow can cross a streamline, and streamlines can never cross one

tube of incompressible aether flow in a vacuum, then from

another. The collection of all streamlines in a region at a given

equation (3.6-14) we obtain:

instant constitutes the instantaneous flow pattern. Note that a streamline describes the velocity field pattern at a given instant of time while a fluid point particle trajectory describes the flow pattern over some finite period of time. !

If the flow is steady (does not vary with time), the flow  velocity υ of fluid point particles will be constant so that at ! each point ∂υ ∂t = 0 . Streamlines will then coincide with fluid point particle trajectories. If the flow is unsteady (varies with  time), the flow velocity υ of fluid point particles will not be

can flow across it, so that this imaginary surface is as impermeable to

!

If a volume element ΔV is taken to be a section of a stream

∫∫

! ! υ • dS = 0 !

ΔS

(3.6-15)

where ΔS is the surface area of this section of stream tube. From equations (3.6-14) and (3.6-12) we see that fluid flow described by equation (3.6-15) is solenoidal. !

If ΔS1 and ΔS2 are the surface areas of the cross-sectional

areas of this stream tube section as shown in Figure 3.6-1, equation (3.6-15) can be written as: 63

!

∫∫

! ! υ • dS = 0 =

ΔS

∫∫

! ! υ • dS +

ΔS1

∫∫

! ! υ • dS !

ΔS2

(3.6-16)

where υ1 is the speed of aether flowing through ΔS1 , and υ 2 is   the speed of aether flowing through ΔS2 . Note υ and dS are in opposite directions for ΔS1 , but in the same direction for ΔS2 . !

Equation (3.6-17) is the equation of continuity of

incompressible flow, and is valid for any stream tube of aether flow in a vacuum not in proximity to a nucleon. From this equation we see that the volume of flowing aether through area

ΔS1 per unit time is the same as the volume of aether flowing through area ΔS2 per unit time. As Maxwell (1855) stated “The quantity of fluid which in unit of time crosses any fixed section of the tube is the same at whatever part of the tube the section be taken.” If the stream tube contracts so that ΔS1 > ΔS2 , we must then have

υ 2 > υ1 . Aether must accelerate, therefore, within the stream tube section between the surface areas ΔS1 and ΔS2 . This is exactly the cause of gravitational acceleration since aether converges into a nucleon. Figure 3.6-1! Fluid flow through a section of a stream tube.

3.7! FORCE DENSITY OF FLOWING AETHER

!

An integral over the sides of the tube is not included on   the right side of equation (3.6-16) since υ is orthogonal to dS

!

for the sides of the tube (by definition of a streamline), and so

steady flowing aether is acting. When the aether in a region is

no flow occurs through the sides. We then have:

flowing in a steady manner, no shear stresses will be acting.  Therefore the force density f must result from normal forces

!

υ1 ΔS1 = υ 2 ΔS2 !

(3.6-17)

We will now consider a small volume element ΔV of  aether upon which a force density f (force per unit volume) of

acting upon the surface area ΔS of the volume element ΔV . 64

Such normal forces can be expressed in terms of the scalar hydrodynamic pressure P : !

∫∫∫

ΔV

! f dV = −

! P dS !

∫∫

ΔS

(3.7-1)

where the area element vector is taken as positive in the outward direction. Using the gradient theorem given in equation (C-62) of Appendix C, we can rewrite equation (3.7-1) as: !

∫∫∫

ΔV

! f dV = −

∫∫∫

! ∇P dV !

!

NATURE OF MECHANICAL FORCE In his Principia, Newton defines an impressed force as “an

action exerted upon a body, in order to change its state, either of rest, or of uniform motion in a straight line.” Generalizing from this definition, we will define (unbalanced) force to be any action on a body that causes the body to accelerate. !

Galileo Galilei (1638) first discovered that a force exerted

upon a body causes the body to accelerate rather than to move (3.7-2)

ΔV

 Since ΔV is arbitrary, the force density f of flowing aether in terms of the hydrodynamic pressure P of aether is given by:   ! (3.7-3) f = − ∇P ! A pressure differential in aether is therefore a force density that, if unbalanced, will cause aether acceleration. !

Since gravitational force results from the steady flow of  aether into matter, gravitational force density f can be expressed in terms of the gradient of the hydrodynamic pressure P of aether using equation (3.7-3). Gravitational force  density f is therefore the negative pressure gradient resulting from the steady flow of aether into matter.

3.8!

with constant velocity. Application of a nonzero resultant force upon a body will always cause a sudden discontinuous change in the acceleration of the body. Any resulting changes in the velocity and position of the body will not be discontinuous, but will represent the integration over time of the acceleration. !

Any pushing action that occurs directly through physical

contact of contiguous real particles and causes a body to accelerate is known as mechanical force. Mechanical force is therefore contact force involving a push action between contiguous real particles. Material particles are, however, never in physical contact with each other, but are always surrounded by aether. “All pieces of matter and all particles are connected together by the ether and by nothing else,” as Lodge (1925a) noted. Therefore mechanical force defined as contact push action between real particles exists only because aether particles can 65

be in physical contact both with other aether particles and with

two bodies, we will therefore mean only that some mechanical

material particles. Material particles act upon each other only

forces are pushing the two bodies closer together.

through (by means of) the aether. When contiguous aether particles that are accelerating cause other aether particles or a

3.9!

material particle to accelerate, this action is mechanical force.

!

!

In twentieth century physics, the concept of force was

(nucleons) being field flow sinks for the nearly incompressible

generally regarded as merely an artificial invention since force

aether. The accelerating flow of aether into a nucleon produces

often appears to serve only as an “intermediate term” between

a negative pressure gradient in the aether surrounding the

two motions (Jammer, 1957). Nevertheless, the changing

nucleon. This negative pressure gradient is gravitational force.

momentum of aether is an action that can cause the acceleration

We see then that gravitational force both results from and

of a body. Defined as an action that causes acceleration, force

causes the acceleration of aether. Gravitational force can be

certainly can be considered to be as real as is any motion.

characterized, therefore, as an action occurring through the

If a force appears to be acting-at-a-distance, the force is

NATURE OF GRAVITATIONAL FORCE Gravitational force is the direct result of material particles

physical contact of contiguous aetherons. We can conclude that

really acting through contiguous aether particles as Descartes

gravitational force is a mechanical force.

proposed. Action-at-a-distance force does not exist. The concept

!

of action-at-a-distance is counterintuitive and has no physical

individual particles of aether do not act over any measurable

validity.

distance. The long-range force known as gravity is produced by

Since aether particles act only upon neighboring particles,

Finally, we note that the nonexistence of any force that

a flow motion involving a series of contiguous aether particles.

acts-at-a-distance has a very important implication: there can be

The flow of the fluid aether into matter is evident to us through

no such thing as a true attractive force. In physics this term

the resulting gravitational pull. Gravitational force is an

should not then be taken to mean force acting-at-a-distance. In

attractive force. Finally, we note that gravitational force is not

this book we will follow Newton (1687) by taking attractive

action-at-a-distance.

force as indicating only the relative direction in which forces

We will designate any force produced by the flowing

are acting. When we say that an attractive force exists between

motion of a fluid as a flow force. All other forces will be 66

designated as non-flow forces. Gravitational force is then a

the surface but also the interior parts of a material body is

flow force. Flow forces are distinguished from non-flow forces

known as a body force.

by the following: 1.! If a non-flow force is accelerating you, it is possible for you to detect this acceleration without reference to any other material bodies. We will reserve the designation impressed force for non-flow forces. 2.! If a flow force is uniformly accelerating you, it is impossible for you to detect this acceleration without reference to other material bodies that are not being so accelerated. You would seem to be weightless. Examples of flow forces acting on a body are: a.! A material body in free fall in a uniform gravitational field. b.! A material body floating beneath the surface of a river whose flow is uniformly accelerating.

3.10! !

SUMMARY

The first mathematical expressions for gravitational force

were derived by Newton and published in 1687 in his Principia. Using Newton’s force law of gravity, Poisson’s field equation of gravity can be derived. From Poisson’s equation we find that a real physical field medium, aether, must exist and be ubiquitous in space, and that some motion must be occurring within this medium to produce the observed gravitational fields. With vector analysis we then determine that matter is a field flow sink for aether. The accelerating flow of aether into matter is manifest as gravitational acceleration. The flow of aether into matter completely explains Galileo’s observation that the gravitational acceleration of a body is independent of the body’s mass and composition. !

Using the concept of matter as a field flow sink for aether,

an aether field equation of gravity can be derived. This field   equation expresses the gravitational field g at a point r in

Because gravitational force results from the action of

space completely in terms of properties of the aether at this

aether particles, such force will penetrate through the surface of

point. Gravity is clearly a fluid dynamic phenomenon of aether.

any body composed of atoms and will affect the entirety of the

Gravitational fields result from mechanical processes occurring

body. A force such as gravity that can act directly upon not only

within the aether as it flows into matter. 67

Newton’s force law of gravity, Poisson’s field equation of gravity, and the aether field equation of gravity are all mutually consistent, and are all valid for a stationary gravitational field of any strength, weak or strong. Einstein’s general theory of relativity does not include a field medium and so is not compatible with any of the gravitational field theories discussed in this chapter (see Massé, 2022 for a detailed evaluation of Einstein’s gravitational theory). !

Finally we note that aether not only serves as the field

medium for gravitational acceleration, but also as the propagation medium for light waves. It can be shown that it is possible to derive Maxwell’s electrodynamic field equations from just a consideration of elastic waves propagating in aether (see Massé, 2021). Aether thus provides the unification of disparate forces and phenomena of nature.

68

Chapter 4 Newtonian Dynamics

! d ! F = (m v ) dt

69

!

The scientific concept of mass has existed for almost four

hundred years. During that time there have been countless

4.1!

studies of mass and numerous revisions in how mass is viewed

!

in physics. Nevertheless, in all that time no clear definition of

quantity of matter, a metaphysical idea developed in the Middle

mass has emerged. This prompted Brown (1960) to state:

Ages (Jammer, 1961). In the seventeenth century, Kepler

“Nothing in the history of science is perhaps so extraordinary as the

introduced the concept of force to provide a cause for planetary

doubt and confusion surrounding the definition of mass.” The

motions and the concept of inertia of matter to describe the

concept of mass has only become more complex through the

resistance of matter to impressed force (see Barbour, 1989).

years: for bodies in motion, mass has appeared to depend on

Newton then proposed that inertia is a material body’s

certain kinematic properties of the bodies; for all material

resistance to a change in motion and that mass is a measure of

bodies, mass has been considered equivalent to energy. The

inertia. He also found that the gravitational force or weight of a

physical reality of mass has even been questioned (Roche,

body is proportional to its quantity of matter and that mass is a

2005).

measure of the quantity of matter. Kepler and Newton thereby

!

began the process of creating the scientific concept of mass.

Another question that has puzzled scientists since the time

CLASSICAL CONCEPTS OF MASS The concept of mass originated from quantitas materiae or

of Kepler is the source of inertia. Because the source of inertia

!

could not be found, some scientists have even appealed to the

mass was generally viewed as real and substantial. All objects

far distant stars for a mechanism to explain inertia.

were thought to consist of mass, and conservation of mass

!

In this chapter, we will briefly review classical concepts of

became a principle of physics. By the twentieth century,

mass. We will then determine the definition and nature of mass.

however, the concept of mass in physics had changed greatly.

We will show how this new definition of mass explains both

Mass was no longer regarded as substantial, but only as a

gravitational mass and inertial mass. We will derive Newton’s

proportionality factor, inertial factor, or a form of energy. Mass

laws of motion from the aether field equation of gravity. Finally,

was often defined then as the quotient of force and acceleration

we will determine the source of inertia.

(Huntington, 1918). Of course, this definition was not without

During most of the eighteenth and nineteenth centuries,

70

its own problems since force as a concept in physics was also

itself but is a measure may have contributed to the difficulty of

being questioned in the twentieth century (Jammer, 1957).

determining the nature of mass. Jackson (1959) stated, “there is no experiment in which mass reveals itself directly.”

4.2!

NATURE OF MASS

!

According to equation (4.2-2), mass exists within a given

To derive a definition of mass, we will consider matter

volume element whenever there is a time variation in the flux

having mass ΔM and density ρ contained within a very small

of aether flowing in the volume element, whether matter is

volume element ΔV of a material body. From equation (3.5-18)

present or not. Equation (2.2-2) is also valid for a volume

we have:

element ΔV that contains only aether. Therefore mass can exist

!

!

1 − 4 π Gζ

∫∫∫

ΔV

∂ ! ! ⎡⎣∇ • (ζ υ ) ⎤⎦ dV = ∂t

∫∫∫

ΔV

ρ dV ! (4.2-1)

!

∫∫∫

ΔV

∂ ! ! ⎡⎣∇ • (ζ υ ) ⎤⎦ dV ! ∂t

thing as matter. This can also be seen from the fact that matter is a field flow sink for aether whereas mass is not. It is matter,

Using equation (3.4-5), we then obtain:

1 ΔM = − 4 π Gζ

without matter. It is evident then that mass is not the same

not mass, that is the cause of gravitational fields. Gravitation is (4.2-2)

not a property of mass, and so mass is not a source of gravitational force. Finally, it is not correct to state that matter

Equation (4.2-2) provides us with a definition of mass from

alone is endowed with mass.

which we can determine the nature of mass. From this equation

!

we see that the mass within a given volume element ΔV is a

of the aether flux (mass) is proportional to the strength of the

measure of the variation with time of the flux of aether in the

sink. The strength of the sink is, in turn, proportional to the

volume element. Therefore mass is a measure of the variation

quantity of matter. It is for this reason that mass is a measure of

with time of the flux of aether. Mass cannot exist without

the quantity of matter in a body as Newton (1687) proposed.

aether. If the time variation of aether flux within ΔV is zero,

Newton’s definition of mass as a measure has puzzled scientists

then the mass within the volume element ΔV is zero.

for a long time (e.g., see Roche, 1988), but we see now that he

!

was correct. Newton never said that mass is the same thing as

As Brown (1960) noted, mass is not an entity that can be

measured; mass is a measure. The fact that mass is not an entity

For aether flowing into a sink (matter), the time variation

matter. 71

!

If ΔS is the surface area of the volume element ΔV , we

can use Gauss’s theorem given in equation (C-58) of Appendix

Since aether flow is incompressible when not in proximity to nucleons, the density of aether is then constant:

C to rewrite equation (4.2-2) as:

∫∫

1 ΔM = − 4 π Gζ

!

ΔS

! ∂ ! ζ υ • d S ( ) ! ∂t

!

∂ζ =0! ∂t

(4.2-6)

!

! ! ∇ζ = 0 !

(4.2-7)

(4.2-3)

This equation shows that the mass within a given volume element is a measure of the variation with time of the momentum density of aether flowing through the surface of the volume element. Note that this is just equation (3.5-1) with

k = ζ . From equation (4.2-3) we see that mass is a scalar that

and so the definition of mass given in equations (4.2-3) and (4.2-5) can be rewritten as: !

can be uniquely specified since it is defined relative to a unique reference frame, aether (see Section 4.7). We also see that mass ! ! is always positive (since υ and dS are oppositely directed). !

The definition of mass given in equation (4.2-2) can be

written in the form: !

ΔM = −

1 4 π Gζ

∫∫∫

ΔV

! ! ⎛ ∂ζ ! ⎞ ⎤ ⎡! ∇ • ζ α ⎢ ( ) + ∇ • ⎜⎝ ∂t υ ⎟⎠ ⎥ dV ! (4.2-4) ⎣ ⎦

! ! where α is ∂υ ∂t (the time variation of the aether flow velocity ! ! υ at a point r ). We then have: ΔM = −

1 4 π Gζ

∫∫∫

ΔV

! ⎛ ∂ζ ! ⎞ ⎤ ⎡ ! ! ! ! ζ ∇ • α + α • ∇ ζ + ∇ •⎜ υ ⎟ ⎥ dV ! (4.2-5) ⎢ ⎝ ⎠⎦ ∂t ⎣

!

∫∫

1 ΔM = − 4π G ΔM = −

ΔS

! ! α • dS !

∫∫∫

1 4π G

ΔV

(4.2-8)

! ! ∇ • α dV !

(4.2-9)

!

Because of the relations given in equations (4.2-6) and ! (4.2-7), α represents the acceleration of gravitational aether ! ! flow when not in proximity to nucleons. Therefore α = g as given in equation (3.5-12), and we can write equation (4.2-9) as: !

∫∫∫

1 ΔM = − 4π G

ΔV

! ! ∇ • g dV !

(4.2-10)

Using equation (3.4-10) we then have: !

ΔM = −

1 4π G

∫∫∫

ΔV

[ − 4 π G ρ ] dV = ∫∫∫

ΔV

ρ dV ! (4.2-11)

which is equation (3.4-5). 72

4.3! !

MASS TYPES The nature of mass is completely specified by the

4.3.1! !

ACTIVE GRAVITATIONAL MASS

! If g is the gravitational field resulting from aether flowing

definition of mass given in equation (4.2-2), and so all mass

into the material body of mass M shown in Figure 3.3-1,

must be consistent with this definition. Therefore only one

equation (3.4-1) applies and we have:

kind of mass exists. While different mass types have been

!

previously identified, we will now show that the designation of a mass type is based, not upon differences in the nature of mass

M ! g = − G 2 rˆ ! r

(4.3-1)

The mass M

itself, but upon differences in the nature or situation of entities

of the material body responsible for the ! gravitational field g is designated as the active gravitational

possessing the mass. We will also see that not all entities are

mass for this field. Active gravitational mass is then a measure

capable of possessing all types of mass; some entities having

of the quantity of matter (the strength of the sink) that is

mass are not such that the mass they possess can ever be

producing the gravitational field. The strength of a sink within

designated active gravitational mass.

a given volume element ΔV is, in turn, proportional to the

!

We will examine three different types of mass that have

variation with time of the flux of aether in ΔV as can be seen

been identified in the scientific literature: active gravitational

from equation (4.2-2). Therefore active gravitational mass is

mass, passive gravitational mass, and inertial mass. To do this,

consistent with the definition of mass given in equations (4.2-2)

we will again consider two bodies of mass M and m ,

and (4.2-3).

respectively (see Figure 3.3-1). In this case, however, we will

!

require only the body of mass M to be material, while the body

gravitational mass, it must contain matter since only matter can

of mass m can be either material or nonmaterial. An example of

create a gravitational field. Gravity is a property of matter, not

a nonmaterial body is the electron (see Massé, 2022)). We will

of mass. The mass of electrons, for example, cannot be

begin by specifying the location in space of the body of mass m ! by the position vector r that has its coordinate system origin at

designated as active gravitational mass since electrons do not

the center of the stationary body of mass M .

gravitational field. The designation of mass as active

For a body to have mass that is designated as active

possess

matter.

Therefore

electrons

do

not

possess

a 73

gravitational mass arises then simply from a consideration of

material, for example, the actual source of the mass m is the

the source of the variation with time of the flux of aether that is

quantity of matter in the body of mass m . Therefore its mass m

mass.

can be designated as both active and passive gravitational

4.3.2!

PASSIVE GRAVITATIONAL MASS

!

The body of mass m shown in Figure 3.3-1 will experience ! acceleration due to the gravitational field g of the material body of mass M . From equation (3.4-1) we have: ! M ! F ! g ≡ = − G 2 rˆ ! (4.3-2) m r ! where F is the gravitational force on the body of mass m . Regardless of whether the body of mass m is a material body or a nonmaterial body, it will experience the same gravitational ! acceleration g since the gravitational field of the body of mass

M is independent of both the composition and mass of the body of mass m . When the body of mass m is experiencing a ! gravitational force due to the gravitational field g of the material body of mass M , the mass m is designated as passive gravitational mass. The designation of passive gravitational mass for m does not arise then from a consideration of the source of the mass m but only from the fact that the body of mass m lies within a gravitational field of some other body. !

The mass m is a measure of the variation with time of the

mass. Both the mass M and the mass m are then defined by equations (4.2-2) and (4.2-3). The designation of passive gravitational mass arises simply from a consideration of the situation within which the body possessing the mass m exists.

4.3.3! !

EQUALITY OF ACTIVE AND PASSIVE GRAVITATIONAL MASS

Since active gravitational mass and passive gravitational

mass are both defined by equations (4.2-2) and (4.2-3), the active and passive gravitational mass of any given material body must not only be equal, but identical. Experiments by Kreuzer (1968) and by Bartlett and Van Buren (1986) have indeed found the active and passive gravitational mass of a material body to be equal. !

For a nonmaterial body, passive gravitational mass is also

defined by equation (4.2-2). A nonmaterial body can never have mass that is designated active gravitational mass, however, since such a body does not contain matter. If a nonmaterial body has mass that is designated passive gravitational mass, the source of this mass must be other than matter.

flux of aether in the body of mass m . If the body of mass m is 74

4.3.4! !

INERTIAL MASS

!

Since the aether is being accelerated by an impressed force

rather than by flowing into a nucleon, aether density will

We will now consider a material body with gravitational

remain constant. Both equations (4.2-6) and (4.2-7) will then

mass m . Using equation (3.4-1), we can rewrite the aether field

always hold. Therefore the term containing the aether flow ! ! ! velocity υ in equation (4.3-4) will always be zero (even if υ ≠ 0 )

equation of gravity (3.5-20) for this mass in the form: ! ! ⎡F⎤ 1 ∂ ! ! ⎡⎣∇ • (ζ υ ) ⎤⎦ = ∇ • ⎢ ⎥ ! ! (4.3-3) ζ ∂t ⎣m⎦ This is basically a field equation describing the relation between ! the motion of aether and the acceleration F m . Whenever

because ∂ζ ∂t = 0 , and we will have: ! ! ! ! ⎡F⎤ ! ∇ •α = ∇ • ⎢ ⎥ ! ⎣m⎦

(4.3-5)

aether is being accelerated, equation (4.3-3) must apply.

and so: !

!

Taking the time derivative in equation (4.3-3), we have: ! 1 ! ⎡ ! ∂ζ ! ⎤ ! ⎡ F ⎤ ! ∇ • ⎢ζ α + υ ⎥ = ∇ • ⎢ ⎥ ! (4.3-4) ζ ∂t ⎦ ⎣ ⎣m⎦ ! There is nothing in equation (4.3-3) restricting F to be only a ! flow force, such as gravitational force. The force F can also be a

!

non-flow force, which is impressed force, and equation (4.3-3)

!

must remain valid. This is evident since impressed force acting

constant since motion of the body will not change the

on a body involves the acceleration of aether. ! ! If F is an unbalanced impressed force acting on the body

gravitational force of the body produced by the matter of the

of mass m , then this force will cause aether to accelerate as

force, however, the gravitational mass m is now designated as

given by equation (4.3-3). The accelerating aether will in turn

inertial mass.

! ! mα = F !

(4.3-6)

! Due to the impressed force F , the aether will have an ! acceleration α . The accelerating aether will cause the material ! ! ! body of mass m to acquire an acceleration a where a = α . Equation (4.3-6) can then be written: ! ! ! F = ma!

(4.3-7)

The body still has its gravitational mass m . This mass is

body. Because the body is being accelerated by an impressed

cause the body of mass m to accelerate. 75

4.3.5! !

EQUALITY OF PASSIVE GRAVITATIONAL MASS AND INERTIAL MASS

4.3.6! !

MASS

We can conclude that, for a material body, active

Clearly gravitational mass and inertial mass are not only

gravitational mass, passive gravitational mass, and inertial

equal, but identical. The equality of gravitational mass and

mass are all equal since they are all simply different

inertial mass is not some incredible coincidence, but follows

designations for the very same mass. For a nonmaterial body,

directly from the aether field equation of gravity. It is then not

passive gravitational mass and inertial mass are equal. A

necessary to assume the equality of passive gravitational mass

nonmaterial body cannot have mass that is designated as active

and inertial mass as has been done in the weak equivalence

gravitational mass.

principle of general relativity. !

The equality of passive gravitational mass and inertial

4.4!

mass for material bodies was first determined experimentally

DERIVATION OF NEWTON’S LAWS OF MOTION

by Newton (1687), and later by Bessel (1832), Southerns (1910),

!

and Potter (1923) using observations of the period (swing time)

a body changes when the body is acted upon by a force.

of pendulums made of different materials. This equality has

Newton’s laws of motion, therefore, are laws pertaining to

been tested to greater precision by torsion balance experiments.

acceleration.

Such experiments were first conducted by von Eötvös (1890)

!

and von Eötvös et al. (1922). These experiments together with

definitive form in his Principia. These laws were developed by

the more recent torsion balance experiments of Dicke (1961a),

Newton from empirical observations. They are the foundation

Roll et al. (1964), Braginsky and Panov (1972), Heckel et al.

for the entire field of mechanics. Newton formulated his laws of

(1989), Adelberger et al. (1990), and Touboul, et al. (2017) show

motion for material point particles. Newtonian mechanics is

that the passive gravitational mass and inertial mass are equal

therefore based upon the theoretical concept of material

15

to within several parts in 10

(see Cook, 1988).

Newton’s three laws of motion describe how the motion of

Newton’s laws of motion appeared for the first time in

particles abstracted to material point particles (which have no volume but do have constant density). Newton’s laws of 76

! ! If an impressed force F ≠ 0 is acting on the point particle

motion can all be derived directly from the aether field

!

equation of gravity.

of mass m , we will have from equation (4.4-1): ! ! ! a ≠ 0!

4.4.1! !

NEWTON’S FIRST LAW OF MOTION

Newton’s first law of motion known as the principle of Every body remains in its state of rest or of uniform motion in a straight line unless it is compelled to change that state by forces impressed upon it. Newton’s first law of motion can be obtained directly from

equation (4.3-7) which was derived using the aether field equation of gravity. We will now consider a material point ! particle of mass m located at a point r . From equation (4.3-7) we have: !

Therefore if a force is impressed on a material point particle, the particle will be compelled to change its state of motion (the

inertia is:

!

(4.4-3)

! ! F = ma!

(4.4-1)

! We will let the force F in equation (4.4-1) be an impressed (non! flow) force acting on the material point particle at point r . ! ! ! If F = 0 so that no force is acting on the material point particle of mass m , we will have from equation (4.4-1): ! ! ! a = 0! (4.4-2) Therefore if no force is impressed on the material point particle, the particle will remain in a state of rest or of uniform rectilinear motion.

particle will accelerate). !

! ! By considering the two cases of impressed force F = 0 and

! ! F ≠ 0 , we have obtained Newton’s first law of motion from

equation (4.3-7). We have shown therefore that Newton’s first law of motion can be derived from the aether field equation of gravity. Newton’s first law of motion provides the qualitative definition of force (Jammer, 1957). !

If there are several forces acting on the material point

particle at the same time, the above discussions will remain ! valid providing that F is taken to be the resultant force (the ! ! vector sum of all forces acting). If the resultant force F = 0 , the forces acting on the material point particle are balanced and are then in equilibrium. The particle will therefore remain in its state of rest or of uniform motion in a straight line. The concept of balanced forces is fundamental to the field of statics in ! ! mechanics. If the resultant force F ≠ 0 , the forces acting on the material point particle are unbalanced, and the particle will be compelled to change its state of motion (accelerate). 77

4.4.2! !

NEWTON’S SECOND LAW OF MOTION

Newton’s second law of motion, which is known as the

fundamental principle of dynamics, is:

!

!

We see therefore that Newton’s second law of motion can

be derived from the aether field equation of gravity. The derivation of equation (4.4-5) given above is consistent with

The change in momentum of a body is proportional to the force impressed upon it and is in the direction of the straight line in which that force is impressed.

Newton’s concept of “the accelerative action of force as a series of

Newton considered the momentum of a body to change in

law of motion, therefore, Newton gave a relation for impulsive

a series of velocity jumps resulting from force impulses. Today, however, his second law of motion is generally written in terms of either acceleration or the time derivative of momentum.

successive actions that imparts to the moving object successive increments of velocity,” as stated by Jammer (1957). In his second forces, not for continuous forces (Cohen, 2002). From this derivation follows the definition: The total impulse of external forces on a body in a given small time interval is equal to the change

from the aether field equation of gravity. To show this, we will

in linear momentum of the body in the same time interval: ! ! ! F Δt = Δ ( m v ) ! (4.4-6)

again consider a material point particle of mass m (as Newton ! did) located at a point r . We can then use equation (4.4-1) in the

4.4.3!

form:

!

!

!

Newton’s second law of motion can be obtained directly

! ! dv ! F = ma = m ! dt

NEWTON’S THIRD LAW OF MOTION

Newton’s third law of motion is: To every action there is always opposed an equal reaction; or the mutual actions of two bodies upon each other are always equal and oppositely directed.

(4.4-4)

! where v is the velocity of the material point particle. Since the density of a material point particle is constant, its mass will be

!

constant. Therefore we can rewrite equation (4.4-4) as:

from Newton’s second law of motion. If we consider a physical

!

! d ! F = [m v ]! dt

(4.4-5)

Newton’s third law of motion can be obtained directly

system consisting of two material point particles having masses ! ! m1 and m2 and velocities v1 and v2 , respectively, and if no

Equation (4.4-5) is Newton’s second law of motion. 78

resultant external force is acting upon this system, we can use

masses, respectively, of body 2. From Newton’s force law of

equation (4.4-5) to write for the system:

gravity as given in equation (3.3-4) we then have:

!

! d ! ! 0 = [ m1 v1 + m2 v2 ] ! dt

(4.4-7)

or !

! m m F12 = − G A1 2 P2 rˆ ! r

!

(4.4-10)

for the gravitational force on body 2 exerted by body 1 and

d d ! ! m1 v1 ] = − [ m2 v2 ] ! [ dt dt

(4.4-8)

Using equation (4.4-5) again, we see from equation (4.4-8) that the force one material point particle of the system exerts on the other material point particle of the system is given by: ! ! ! F1 = − F2 ! (4.4-9)

! m m F21 = G A2 2 P1 rˆ ! r

!

(4.4-11)

for the gravitational force on body 1 exerted by body 2 (where we are using the same unit vector rˆ in both equations). From Newton’s third law of motion we can write: !

which is Newton’s third law of motion for two material point

! ! m m m m F12 = − G A1 2 P2 rˆ = − F21 = − G A2 2 P1 rˆ ! r r

(4.4-12)

mA1 mA2 ! = mP1 mP2

(4.4-13)

particles interacting with each other. Therefore Newton’s third

and so:

law of motion follows from the aether field equation of gravity. ! ! Note that F1 and F2 do not act on the same point particle but on

!

two different point particles. Also note that both forces must be

which must hold for any two material bodies regardless of

acting simultaneously for this law to be valid.

composition. These ratios must then equal a constant (which

!

can be chosen to be one) since they are the same for bodies of

The proportionality of active and passive gravitational

mass is easily demonstrated from Newton’s third law of

different compositions and weights.

motion. We will consider two material bodies that are stationary relative to each other. We will let mA1 and mP1 be the

4.4.4!

ORIGIN OF NEWTON’S LAWS OF MOTION

active and passive gravitational masses, respectively, of body 1

!

and mA2 and mP2 be the active and passive gravitational

in and can be derived from the aether field equation of gravity.

To summarize, Newton’s laws of motion are all contained

79

The origin of Newton’s three laws of motion can all be found then in the motion of aether. This means that the foundation of

4.5.1!

Newtonian dynamics is to be found in aether field theory. Newton’s laws of motion are valid only because of the

If the resultant impressed force on a body of mass m is   F = 0 , we have from Newton’s second law of motion given in

existence and properties of aether. Moreover, Newton’s laws

equation (4.4-5):

of motion are valid only with respect to inertial frames of reference (see Section 4.7).

4.4.5!

EQUATION OF MOTION

! An equation for the acceleration a of a body as a function ! ! of r , v , and t is called the equation of motion of the body. In !

CONSERVATION OF LINEAR MOMENTUM

d ! ! mv] = 0! [ dt

!

! and so the momentum p is given by: ! ! ! p = m v = constant vector !

(4.5-1)

(4.5-2)

Therefore if no resultant force is acting on a body of mass m , its

terms of rectangular components an equation of motion has the

momentum will be a constant vector. This is known as the law

form:

of conservation of linear momentum or simply as the law of

!! ! t)! x = !! x ( x, x,

!

(4.4-14)

The motion of the body is then obtained by determining its position as a function of time x ( t ) .

4.5! !

DERIVATION OF CONSERVATION LAWS We will now derive a number of important conservation

laws. A physical entity is considered to be conserved if it is invariant (does not change) with time.

conservation of momentum. Since this law was derived using Newton’s second law of motion, conservation of momentum is limited to inertial frames of reference (see Section 4.7).

4.5.2! !

CONSERVATION OF ANGULAR MOMENTUM

! ! ! When m is a constant and since v × v = 0 , we can use

equation (4.5-1) to write: !

! d ! d ! ! ! r × [ m v ] = ⎡⎣ m ( r × v ) ⎤⎦ = 0 ! dt dt

(4.5-3)

! where r is the position vector for the body of mass m . We then have: 80

! ! ! ! ! ! m ( r × v ) = r × ( m v ) = r × p = constant vector !

!

(4.5-4)

Therefore if no resultant torque is acting on a body of constant ! ! mass m , its angular momentum r × p will be a constant vector. This is known as the law of conservation of angular ! momentum. In a central force field, the linear momentum p is ! ! ! ! in the same direction as r and so we have r × p = 0 . Angular momentum is always conserved then in a central force field.

4.5.3! !

CONSERVATION OF ENERGY

! ! When there exists a force F ≠ 0 that is derivable from a

scalar potential V so that: ! ! ! F = − ∇V ! (4.5-5) ! ! the force F is known as a conservative force if F does not ! depend on time. If F is acting along some path C , we have: !

∫

C

! ! F • dr = −

∫

! ! ∇V • dr = −

C

∫ dV ! C

(4.5-6)

Using Newton’s second law of motion for a material point particle of mass m , we can rewrite equation (4.5-6) as: ! d ! dr ! ( m v ) • dt dt = − dV ! (4.5-7) C dt C

∫

or

∫

!

m

∫

C

! dv ! • v dt = − dt

∫ dV !

(4.5-8)

∫ dV !

(4.5-9)

C

Therefore !

1 m 2

∫

T=

1 ! ! 1 m v i v = m v2 ! 2 2

C

! ! d (v • v ) = −

C

Letting !

(4.5-10)

We can rewrite equation (4.5-9) as: !

∫ dT = − ∫ dV ! C

(4.5-11)

C

where T is defined as the kinetic energy and V is defined as the potential energy of the point particle. Note that momentum ! ! p = m v and kinetic energy T are not independent since ! T = p 2 2 m where p is the magnitude of p . ! !

Letting E = T + V , from equation (4.5-11) we have:

dE d (T + V ) = = 0! dt dt

(4.5-12)

E = T + V = constant !

(4.5-13)

or !

which is known as the law of conservation of energy. 81

!

If the force is a function of time so that the force is not

conservative, then energy will not be conserved. We can show this using E = T + V : !

dE dT dV = + ! dt dt dt

(4.5-14)

! where V is a function of position and time: V = V ( r, t ) . We can rewrite this equation using equation (4.5-10): ! ! dE ! d v ∂V d r ∂V ! ! = mv • + ! + dt dt ∂r dt ∂t

dE ! ! ! ! ∂V ! = F • v + ∇V • v + dt ∂t

(4.5-15)

dE ! ! ! ! ∂V ∂V = F • v − F• v + = ≠ 0! dt ∂t ∂t

(4.5-18)

Using equation (4.5-5), we can write: ! ! ! − ∇V = m ∇ϕ !

(4.5-19)

V = − mϕ !

!

(4.5-20)

where ϕ is the gravitational potential of a gravitational field and V is the gravitational potential energy in the field of a

(4.5-16)

material point particle having mass m .

4.5.4!

Using equation (4.5-5): !

for a material point particle of mass m : ! ! ! ! F = m g = m ∇ϕ !

or

or !

gravitational field. From equations (3.4-1) and (3.4-13) we have

(4.5-17)

!

FUNDAMENTAL JUSTIFICATION FOR CONSERVATION LAWS

The three conservation laws: conservation of linear

and so energy is not conserved.

momentum,

conservation

of

angular

momentum,

and

!

The potential energy V in equation (4.5-5) should not be

conservation of energy can all be obtained from Newton’s laws

confused with the gravitational potential ϕ , which was defined

of motion. Therefore these conservation laws all derive

in equation (3.4-11); potential energy and gravitational potential

indirectly from the aether field equation of gravity. These

have different physical dimensions and refer to different

conservation laws all have their origin then in the motion of

entities. Gravitational potential characterizes a gravitational

aether, and properties of aether provide the fundamental

field while gravitational potential energy refers to the energy a

justification for the conservation laws. Or as J. J. Thomson

given body has by being in a certain position within a

(1904a) said, “all mass is mass of the ether, all momentum, 82

momentum of the ether, and all kinetic energy, kinetic energy of the

or using equation (4.5-10):

ether.” For this reason the three conservation laws: !

1.! Conservation of momentum.

∫

C

2.! Conservation of angular momentum.

d ⎡1 2⎤ m v ⎥⎦ dt = dt ⎢⎣ 2

∫

C

dT dt = dt

∫

! ! F • v dt =

C

∫

dW dt ! (4.5-23) C dt

and so we have:

3.! Conservation of energy. have not only an empirical basis, but also a theoretical basis (applying equally at macroscopic and subatomic levels).

dT ! ! dW ! = F•v = dt dt

!

(4.5-24)

Therefore the rate at which a conservative force field moving a material particle does work is equal to the time variation of the

4.5.5! !

WORK

! If F is a conservative force field, then for a material

particle’s kinetic energy.

particle of mass m moving along some path C in the field, we

4.6!

can rewrite equation (4.5-6) as:

!

∫

!

! ! F • dr = −

C

∫

! ! ∇V • dr = −

C

∫ dV = ∫ dW ! C

C

(4.5-21)

moving it around any closed path is zero. For a material particle of mass m , we can use equations

(4.5-9) and (4.5-7) to write:

1 m 2

∫

C

! ! d (v • v ) =

∫

C

! d ! dr ( mv ) • dt = dt dt

From his study of spherical bodies moving on inclined

planes, Galileo Galilei (1638) determined that: “any velocity once

where dW is defined as the differential work done by the force ! ! F acting over a distance dr in the force field. From equation ! (4.5-21) we see that the net work done by F on a particle in !

NATURE OF INERTIA

imparted to a moving body will be rigidly maintained as long as the external causes of acceleration or retardation are removed.” Galileo thought, however, that the velocity would remain constant only for horizontal motion along the Earth’s surface (where gravity could not provide horizontal acceleration). Descartes (1644) stated that: “each thing, as far as is in its power, always remains in the same state; and that consequently, when it is once moved, it always continues to move,” and that: “all movement is, of itself,

∫

C

! ! F • v dt =

∫

C

dW ! (4.5-22)

along straight lines.” Descartes was the first to realize the more general law: a body will remain in a state of rest or of uniform 83

rectilinear motion unless acted upon by an external force. This

origin of inertia is and remains the most obscure subject in the theory

law is known today as Galileo’s law of inertia.

of particles and fields.”

!

Newton defined inertia to be a measure of the resistance of

a body to any change being made in its state of motion. A

4.6.1!

change in its state of motion is, of course, acceleration. A body

!

in a state of rest or of uniform rectilinear motion will then not

aether field equation of gravity. From the definition of inertia ! we see that a body of mass m located at a point r will possess

have inertia since the body is not changing its state of motion (accelerating). Because a body cannot be accelerated without a force acting on the body, a body will possess inertia only if a force is acting on the body. Moreover, since resistance is involved for inertia to be present, the force must be a non-flow force. Such a force will then be an impressed force. Inertia can be defined, therefore, as a measure of the resistance of a body to any change in its state of motion due to an impressed force.

DERIVATION OF GALILEO’S LAW OF INERTIA

It is possible to derive Galileo’s law of inertia from the

inertia only if a nonzero impressed force: ! ! ! F ≠ 0!

is acting to accelerate this body by causing the aether to accelerate so that: ! F ! ! ≠ 0! m

The concept of inertia is obviously associated with Newton’s

!

laws of motion and, in particular, with Newton’s first law of

(4.3-3) that we must have:

motion (which is also referred to as Newton’s principle of inertia). !

Inertia is a fundamental concept in physics. Nevertheless

the source of inertia has remained a mystery ever since Kepler (1620) first associated inertia with matter almost four centuries ago. As Feynman (1965) noted, “The reason why things coast for

!

(4.6-2)

For equation (4.6-2) to be true, we see from equation

! ! 1 ∂ ! F ⎡ ⎤ ! ⎡⎣∇ • (ζ υ ) ⎤⎦ = ∇ • ⎢ ⎥ ≠ 0 ! ζ ∂t ⎣m⎦

(4.6-3)

Equation (4.6-3) is the necessary condition for a body to possess inertia. Taking the time derivative we have: !

ever has never been found out. The law of inertia has no known origin.” Or as Pais (1982) stated, “It must also be said that the

(4.6-1)

1 ! ⎡ ! ∂ζ ! ⎤ ∇ • ⎢ζ α + υ ⎥ ≠ 0 ! ζ ∂t ⎦ ⎣

(4.6-4)

or 84

!

! ∂ζ ! ζ α + υ ≠ 0! ∂t

! (4.6-5)

!

From equation (4.6-5) we see that the necessary condition ! for a body at any point r to possess inertia can be expressed completely in terms of properties of the aether at this same ! point r . The origin of inertia must then be found in the physical properties of aether. This explains how it is that inertia can be present instantaneously wherever an impressed force acts on a body. Aether is ubiquitous in our Universe. !

From equation (4.6-5) we also see that a body can have

inertia only if at least one of the two terms of this equation is nonzero. Since the density of aether ζ is constant for aether flow not in proximity to nucleons, we have ∂ζ ∂t = 0 and so, for aether motion resulting from an impressed force, the second ! term will always be zero regardless of the value of υ : !

∂ζ ! ! υ = 0! ∂t

(4.6-6)

! ! ! For inertia to exist, therefore, we must α ≠ 0 . Any body at ! ! rest or moving at a constant velocity (so that α = 0 ) will not

We can now understand why a body moving at a constant

velocity does not have inertia, but an accelerating body does. We can also see why uniform motion through space does not produce observable effects, while accelerated motion does. !

We have found that Galileo’s law of inertia can be derived

from the aether field equation of gravity. For the first time we have an explanation why, for inertia to be present in a body, a fundamental

distinction

exists

between

accelerated

and

unaccelerated motion.

4.6.2! !

MACH’S PRINCIPLE

Mach (1883) thought that a material body has inertia only

because it interacts physically in some manner with all the other material bodies in the Universe. Mach regarded “all masses as related to each other.” Mach’s principle (as named by Einstein, 1918b) proposes that all matter is coupled together so that inertia has its physical origin in the totality of the matter of the Universe. Since most matter is located in far distant ‘fixed stars’, Mach concluded that the accelerated motion of a body relative to the far distant stars must be the primary cause of the

have inertia since both terms of equation (4.6-5) will then be

inertia of the body. Why this might be so has never been

zero. An impressed force must be acting to accelerate a body for

explained.

the body to have inertia.

!

Mach’s principle has the serious problem of requiring

instantaneous action-at-a-distance between a material body and 85

the far distant stars. Furthermore as Bondi (1952) noted, it has

describing parameters of the physical entities interacting with

not been possible to express Mach’s principle in mathematical

the reference frame.

form, nor has it been possible to verify it experimentally.

!

Mach’s principle predicts that the inertia of any given material

in motion in an attempt to explain why the rotation of the Earth

body will increase as the amount of matter in proximity to the

had not been detected by observations on Earth (in his day). He

body increases. No such increase in inertia has ever been

determined that: “motion, in so far as it is and acts as motion, to

observed, however (see Hughes et al., 1960; and Drever, 1960).

that extent exists relatively to things that lack it; and among things

!

From equation (4.6-3) we see that Mach’s principle cannot

which all share equally in any motion, it does not act, and is as if it

be correct. The inertia of a body is not directly dependent upon

did not exist.” Galileo found that the velocity of a material body

any other body.

is always relative to the velocity of the reference frame chosen.

4.7! !

ABSOLUTE MOTIONS AND GALILEAN RELATIVITY Spatial position and motion can only be defined relative

Galileo Galilei (1632) investigated the relativity of bodies

The velocity of a material body is not uniquely specified until the frame of reference is also specified. This frame of reference can be any other physical entity. !

The acceleration of a material body must also be relative to

to some real physical entity. Such a physical entity is known as

a reference frame since acceleration is motion. The acceleration

a reference frame or a frame of reference. Coordinate systems

of a material body is found to be absolute, however. A

are geometrical definitions or concepts that are extremely

kinematic absolute designates a kinematic entity that is the

useful mathematical constructions for describing physical

same for all observers everywhere, and therefore a kinematic

entities. Coordinate systems are not physical entities in

absolute must always be relative to a unique reference frame. A

themselves, however. A coordinate system cannot then be a

unique reference frame must then exist for all accelerations.

reference frame. Moreover, a coordinate system cannot interact

!

with physical fields and forces. A coordinate system can, of

considered an absolute reference frame for that motion. We see

course, be placed within a frame of reference for use in

then that the designation of absolute or relative for motion is

A unique reference frame for any given motion is

86

based upon the existence or nonexistence of a unique frame of reference. !

The fact that acceleration of a material body is absolute,

while velocity of a material body is relative is now known as Galilean relativity or Newtonian relativity. From Newton’s first law of motion we can conclude that, if the resultant force on a body is nonzero, the body must be accelerating relative to the absolute reference frame for acceleration. !

A material reference frame that is at rest or is moving with

a uniform rectilinear motion with respect to the absolute reference frame for acceleration is referred to as an inertial reference frame (Lange, 1886). In other words, a material reference frame that is not accelerating is an inertial reference frame. Therefore an accelerating or rotating material body cannot be an inertial reference frame. By simply changing the inertial reference frame being used, a material body’s velocity

Figure 4.7-1! A reference frame S’ is moving with a constant ! velocity U relative to an inertial reference frame S.

can change, but it will still have zero acceleration. !

By considering the transformation laws that allow us to

pass from a coordinate system within one material reference

!

We will consider two reference frames S and S’ that were

co-located and that had identical coordinate system origins and

frame that is moving relative to the first reference frame, we can

orientations at time t = 0 . The reference frame S’ is moving with ! a constant velocity U with respect to the reference frame S,

demonstrate Galilean relativity. We will now consider two such

which is at rest relative to the absolute reference frame for

material reference frames S and S’ indicated by the xi and xj′

accelerations. Both reference frames S and S’ are therefore

coordinate systems, respectively, in Figure 4.7-1.

inertial. A material particle of mass m is referred to both

frame to a coordinate system within a second material reference

87

reference frames. The position of the material particle of mass ! m is given by position vector r with respect to the reference ! frame S, and by position vector r ′ with respect to the reference

(by simply changing the reference frame). If a non-flow force is

frame S’, where: ! ! ! ! r′ = r − U t !

relative to other material bodies (see Section 3.9). (4.7-1)

The three component equations represented by the vector equation (4.7-1) are known as Galilean transformations. In Galilean transformations, time intervals are taken to be absolute. Therefore time is the same in reference frames S and S’. Galilean transformations are valid between inertial reference frames. From equation (4.7-1), the velocity of the material particle is: !

! ! dr ′ dr ! = −U ! dt dt

(4.7-2)

From this equation, the acceleration of the material particle is: ! !

2!

d r′ d r = 2! dt 2 dt

acceleration without any consideration of the body’s motion !

From equation (4.7-2) we see that velocities with respect to

the inertial reference frames S and S’ are not equal, and so velocity is not absolute, but relative to the velocity of the reference frame. Therefore we cannot determine an absolute velocity for a material particle. As Davies and Gribbin (1992) noted, “The contrast between uniform and nonuniform motion is deep.” !

Similarly, we see from equation (4.7-1) that position is not

absolute, but relative to the position of the reference frame. We then cannot determine an absolute position for a material

This is known as the Galileo addition theorem for velocities. 2!

accelerating a material body, it is possible to detect this

(4.7-3)

particle. !

Newton’s laws of motion contain only accelerations, not

velocities. In fact, “Newton’s laws of motion are statements about acceleration,” as Clotfelter (1970) observed. Newton’s laws of motion are invariant with respect to Galilean transformations therefore. This means that a law in mechanics that is valid for

We see from equation (4.7-3) that accelerations with

one inertial reference frame is also valid for any other reference

respect to the inertial reference frames S and S’ are equal, and

frame moving with constant velocity relative to the first frame.

so acceleration is absolute. If acceleration were not absolute,

Relative to inertial reference frames, laws of mechanics take the

then a body could accelerate without the application of a force

form of Newton's laws of motion. 88

Since the acceleration of a material body is absolute,

!

The nonmaterial absolute reference frame for accelerations

implicit in Newton’s laws of motion is the assumption that an

has to be a physical entity with physical properties so that the

absolute reference frame exists relative to which these laws are

motion of material bodies relative to this reference frame can be

valid. Newton’s laws of motion are only valid relative to the

determined. Aether is a real physical entity that is nonmaterial

absolute reference frame or to inertial reference frames.

but, nevertheless, has real physical properties. Moreover aether

Newton’s laws of motion also require the concept of

is ubiquitous and permeates all space. We will now show that

absolute time as was noted by Hawking (1987). There certainly

aether does indeed serve as the absolute reference frame for

are no kinematic reasons for assuming that time is in any way

accelerations.

dependent on motion. Moreover the fact that acceleration is

!

absolute for all inertial reference frames argues against any

equation (4.3-4) is:

dependence of time on motion.

! 1 ! ⎡ ! ∂ζ ! ⎤ ! ⎡ F ⎤ ! ∇ • ⎢ζ α + υ ⎥ = ∇ • ⎢ ⎥ ! (4.7-4) ζ ∂t ⎦ ⎣ ⎣m⎦ ! where we will now consider F to be a non-flow (impressed)

!

Newton struggled with the problem of absolute versus

relative motion, and he tried to find the absolute reference frame for accelerations. He knew that: “True motion is neither generated nor altered, but by some force impressed upon the body moved; but relative motion may be generated or altered without any force impressed upon the body.” Newton realized that no material body may really be at rest and that: “the true and absolute motion of a body cannot be determined by the translation of it from those which only seem to rest.” Therefore he sought an absolute reference frame for acceleration that was nonmaterial. He thought that empty space might provide this absolute reference frame. Empty space has no physical properties, however, and so cannot be a reference frame.

The aether field equation of gravity written in the form of

force. We then have ∂ζ ∂t = 0 since the aether not in proximity to nucleons is incompressible, and so equation (4.7-4) will hold ! ! true for any value of υ , but for only one value of α . We now see how a reference frame can exist that is absolute for acceleration, but is not absolute for velocity. The absolute reference frame for acceleration must be aether. This is confirmed by the fact that Newton’s three laws of motion (which pertain to the acceleration of bodies) can all be derived from a consideration of the motion of aether. Since there is no absolute reference frame for velocity, the velocity of a body is 89

relative to any given inertial reference frame (i.e., any reference frame that has uniform motion relative to the absolute reference frame, aether). We can now understand why inertial reference frames are distinguishable from accelerating material reference frames. Aether is the special, preferential, or privileged reference frame for accelerating bodies. Aether is the absolute reference frame

The motion of a material particle can be specified using ! Newton’s laws of motion. If r is the position vector to a material particle, then the equation of motion of the particle ! ! can be expressed in terms of its acceleration "" r , velocity r" , ! displacement r , and time t : ! ! !" ! "" ! r = "" r ( t, r, r ) ! (4.8-1)

relative to which Newton’s laws of motion are valid. If aether

! ! From the equation of motion, the trajectory r = r ( t ) of the

did not exist, no absolute reference frame for accelerations

material particle can be obtained by integrating equation

would exist.

(4.8-1).

Finally, the fact that someone in free fall in a uniform gravitational field does not sense g-force acceleration can now

4.9!

SUMMARY

be explained. The aether surrounding a person in free fall is

!

also being accelerated (causing the gravitational field), and so

the flux of aether. All mass is consistent with this definition.

there exists no acceleration of the person relative to the absolute

There is only one kind of mass. Active gravitational mass,

reference frame.

passive gravitational mass, and inertial mass are designations

!

The conservation laws of linear and angular momentum

based, not upon differences in the nature of mass itself, but

and of energy are all functions of velocity (see Section 4.5).

upon differences relating to the source of the mass or to the

Therefore these conservation laws are valid only for inertial

situation of an entity possessing the mass.

reference frames.

!

Mass can be defined as a measure of the time variation of

A material body can have mass that is designated as active

gravitational mass, passive gravitational mass, or inertial mass.

4.8! !

NEWTONIAN DYNAMICS Dynamics is the branch of mechanics concerned with the

A nonmaterial body can only have mass that is designated as passive gravitational mass or inertial mass.

motion of material objects as described by the laws of physics. 90

!

Newton’s three laws of motion can all be derived from the

aether field equation of gravity. The theorems for conservation of momentum and conservation of energy can, in turn, be obtained from Newton’s laws of motion. Therefore Newton’s laws of motion and the laws of conservation of momentum and conservation of energy all have their origin in the motion of aether. The foundation of Newtonian dynamics is to be found in aether field theory. !

The inertia of a material body is independent of any other

material body. Mach’s principle is not correct. Aether is the absolute reference frame for all accelerating bodies.

91

Chapter 5 Lagrangian Dynamics

∂L d ∂L − =0 ∂xi dt ∂ x!i

92

!

In this chapter we will present the Lagrangian formulation

of Newtonian dynamics. We will derive the Euler-Lagrange equation and Lagrange’s equations of motion from Newton’s laws of motion and the law of conservation of energy. Since in

!

We can write the expression for kinetic energy given in ! equation (4.5-10) in terms of the position vector r as: !

T=

1 ! ! m ( r" • r" ) ! 2

(5.1-4)

this book we are examining the foundations of the principle of

From this equation we see that kinetic energy is a scalar. In

least action, and not exploring the many uses of Lagrangian

rectangular component form we then have:

dynamics, we will develop all equations using only rectangular coordinates.

!

5.1! EULER-LAGRANGE EQUATION !

Newton’s second law of motion for a material point

! d ! ! F = ( m r" ) = m "" r! dt

3

∑ x!

2

i

!

(5.1-5)

i =1

and so:

particle of mass m can be written in terms of the position vector ! r as: !

1 T= m 2

(5.1-1)

!

dT =m dt

3

∑ x! !!x ! i

i

(5.1-6)

i =1

and so for each component:

d ∂T = m !! xi ! ! dt ∂ xi

! where the mass m is taken to be constant. If F is a conservative

!

force field, we can use equation (4.5-5) to write: ! ! ! "" ! F = m r = − ∇V !

From equations (5.1-7) and (5.1-3) we obtain the three (5.1-2)

From equations (5.1-1) and (5.1-2) in rectangular component form we then have: !

m !! xi = −

∂V ! ∂xi

i = 1, 2, 3 !

(5.1-7)

equations: !

∂V d ∂T + = 0! ∂xi dt ∂ x!i

(5.1-8)

(5.1-3) 93

5.1.1! !

EULER-LAGRANGE EQUATION

Using relations for kinetic energy and potential energy, the

Euler-Lagrange equation can be derived. We are assuming that:

∂T = 0! ∂xi

∂V =0! ∂ x!i

∂( T − V ) d ∂( T − V ) − = 0! ∂xi dt ∂ x!i

(5.1-9)

(5.1-10)

We can rewrite equation (5.1-10) as:

Newtonian dynamics for a stationary conservative force field, and so Lagrangian dynamics can be viewed as simply a !

The component equations represented by the Euler-

is one equation for each of the independent variables xi . They form a set of simultaneous differential equations that describe the motion of the system. The solutions of these equations yield the trajectory of the system. The dynamics of a physical system

(5.1-11)

function of xi , the Lagrangian is a function of x!i , xi , and t :

L = L ( xi ( t ) , x!i ( t ) , t ) !

energy. Lagrangian dynamics is completely equivalent to

Lagrange equation are Lagrange’s equations of motion. There

From equation (1.2-4) we have the Lagrangian:

L = T −V !

Therefore the Euler-Lagrange equation can be derived

reformulation of Newtonian dynamics.

The Lagrangian is a scalar. Since T is a function of x!i and V is a !

(2.6-3). from Newton’s laws of motion and the law of conservation of

Using these equations, we can write equation (5.1-8) as:

!

which is the Euler-Lagrange equation as was given in equation

2.! The force field is conservative.

independent of x!i . Therefore we have:

!

(5.1-13)

!

from equation (5.1-3) we see that for a conservative system V is

!

i = 1, 2, 3 !

1.! The mass of the particle is constant.

From equation (5.1-5) we see that T is independent of xi , and

!

∂L d ∂L − = 0! ∂xi dt ∂ x!i

!

(5.1-12)

are then described by the Euler-Lagrange equation. Note that the Lagrangian is an explicit function of only xi ( t ) and x!i ( t ) , but this is sufficient to completely determine the state of the system. !

The Euler-Lagrange equation is a second-order differential

equation to be solved together with given initial or boundary conditions. One very important advantage of the Lagrangian 94

equations of motion over the Newtonian equations of motion is

pi = m x!i =

!

that the Euler-Lagrange equations are not vector equations. Both kinetic energy and potential energy are scalars. The equivalent Newtonian equations of motion are vector equations since acceleration is a vector. It is much simpler to change coordinate systems in the process of solving a problem if Lagrangian

dynamics

also

differ

from

Newtonian

(

)

(5.1-14)

where T is the kinetic energy of the particle. Since:

∂V = 0! ! ∂ xi

!

(5.1-15)

and L = T − V we can also write equation (5.1-14) as:

vectors are not involved. !

1 ∂ ∂T m x!i2 = ! 2 ∂t ∂ x!i

pi =

!

∂L ! ∂ x!i

(5.1-16)

dynamics in another way. Newton’s equations of motion are expressed in terms of point vectors that pertain to a specific point in space, whereas the action integral from which the Lagrangian equations of motion are derived involves the integration of energy over an interval of time. !

The dynamics of a physical system are defined by the

5.1.3! !

is simply a mathematical ambiguity that does not change the physics of the system.

5.1.2! !

MOMENTUM

! The components of the momentum p of a material particle

of mass m in a rectangular coordinate system are:

If a particular coordinate xi does not explicitly appear in

the Lagrangian, then from equation (5.1-13) we have: !

Lagrangian L = T − V . While it is true that any constant can be added to L without changing the Euler-Lagrange equation, this

NO EXPLICIT X DEPENDENCE

d ∂L = 0! dt ∂ x!i

(5.1-17)

or from equation (5.1-16): !

dpi = 0! dt

(5.1-18)

Therefore we have: !

pi = C !

(5.1-19)

where C is a constant. We therefore obtain conservation of momentum. If a particular coordinate xi does not explicitly 95

appear in the Lagrangian, then xi is called a cyclic coordinate

!

or an ignorable coordinate. We see from the above equations that the component of momentum associated with a cyclic coordinate will always be a constant.

5.1.4! !

or !

BELTRAMI IDENTITY

If the Lagrangian does not depend explicitly on the

independent variable t , we can use the Beltrami identity given

L−

!

∑ i =1

∂L x!i = C ! ∂ x!i

∂T ∂V d ∂T d ∂V − − + =0 ∂xi ∂xi dt ∂ x!i dt ∂ x!i

Since !

in equation (2.3-44) to write: 3

∂L d ∂L ∂(T − V ) d ∂(T − V ) − = − =0 ∂xi dt ∂ x!i ∂xi dt ∂ x!i

∂T =0! ∂xi

we have: (5.1-20)

where C is a constant. Example 5.1-1

!

∂V d ∂T + =0 ∂xi dt ∂ x!i

where !

dT = dt

Show that Lagrange’s equations of motion are equivalent to Newton’s second law of motion for a particle of mass m .

and so:

Solution:

!

For a particle of mass m , the kinetic energy is: !

1 T = m x!i 2 2

The Euler-Lagrange equation is then:

dV =0 dx!i

3

∑

m x!i !! xi !

i =1

m !! xi = −

d ∂T = m !! xi dt ∂ x!i

∂V ∂xi

In vector form this is: ! ! ! ∂V ! F = m "" r = − ! = − ∇V ∂r

96

which is Newton’s second law of motion. So Lagrange’s

or

equations of motion are equivalent to Newton’s laws of

!

motion for a particle. The above solution is simply the reverse of the process used to derive the Euler-Lagrange equation

!! x=−

k x m

which is the equation of motion for the spring.

from Newton’s laws of motion.

5.2! DEGREES OF FREEDOM Example 5.1-2

!

Determine the equation of motion for a spring where:

describe the configuration of a physical system is known as the

!

T=

1 m x! 2 ! 2

V=

The number of independent variables necessary to

degrees of freedom of the system. A material particle that is

1 2 kx 2

free of any contraint will, for example, will require three coordinates to describe its position, and so it has three degrees

Solution: !

The Euler-Lagrange equation is:

!

∂L d ∂L ∂(T − V ) d ∂(T − V ) − = − =0 ∂x dt ∂ x! ∂x dt ∂ x!

of freedom. A physical system consisting of N particles will generally have 3N degrees of freedom. !

Any contraint imposed upon a physical system will

reduce the system’s degrees of freedom. Each contraint reduces

Since

the degrees of freedom by one. The number of degrees of

!

dT = m x! !! x! dt

d ∂T = m !! x! dt ∂ x!

∂T =0 ∂x

freedom that a system of N particles has is then equal to

!

∂V = 0! ∂ x!

d ∂V =0! dt ∂ x!

dV = kx dx

!

and so the Euler-Lagrange equation gives us: !

3N − M where M is the number of contraints. If the relation between the constraints in a coordinate

system has the form: !

φ ( q1, q2 , q3 , !, qn , t ) = 0 !

(5.2-1)

m !! x+kx=0 97

then the constraints are called holonomic, and the physical system is referred to as a holonomic system. From equation (5.2-1) we see that for such a holonomic system one coordinate can be expressed in terms of all the coordinates. Therefore the number of independent coordinates is reduced by one. Example 5.2-1 Determine the degrees of freedom and the equation of motion for a pendulum of length b and mass m . Solution: We will use plane polar coordinates ( r, θ ) . The pendulum is constrained to move in a plane, and r is constrained to be the constant b (see Figure 5.2-1). The number of degrees of freedom is then 3 − 2 = 1 . The pendulum system can therefore be represented by one

Figure 5.2-1! Simple pendulum. The potential energy V is: !

coordinate θ . We have:

V = m gb (1− cosθ )

!

x = b sin θ !

y = − b cosθ

The Lagrangian L is then:

!

x! = b θ! cosθ !

y! = b θ! sin θ

!

The kinetic energy T is: !

(

)

(

)

1 1 1 T = m x! 2 + y! 2 = m b 2 θ! 2 sin 2 θ + cos 2 θ = m b 2 θ! 2 2 2 2

L = T −V =

1 m b 2θ! 2 − m gb (1− cosθ ) 2

Using the Euler-Lagrange equation with the coordinate θ : !

∂L d ∂L d − = 0 = − m gb sin θ − m b 2θ! ∂θ dt ∂θ! dt 98

or !

b θ!! + gsin θ = 0

The equation of motion of the pendulum is then: !

g θ!! = − sin θ b

5.3! SUMMARY !

The Euler-Lagrange equation and Lagrange’s equations of

motion can be derived from Newton’s laws of motion and the law of conservation of energy. Lagrangian dynamics can be considered an alternative formulation of Newtonian dynamics.

99

Chapter 6 Foundations of the Principle of Least Action

S=

∫

L dt =

∫

mv ds − C

100

!

In this chapter we will show that Hamilton’s principle and

the principle of least action of Maupertuis are equivalent in a stationary conservative force field. We will then determine the physical foundation of the principle of least action.

!

S=

!

We will now consider the Lagrangian for a physical

Zeldovich and Myškis (1976), the Lagrangian can be written as: (6.1-1)

where T is the kinetic energy of the system and V is the

∫

t2

E dt = E

t1

system within a stationary conservative force field. As noted by

(6.1-2)

we have: !

!

S=2

(6.1-3)

We can now write the action of the system as defined by

S=

∫ L dt = ∫ 2T dt − ∫ E dt !

For a given time interval [ t1, t 2 ] we have:

t2

∫ dt = E (t − t ) = C ! 2

(6.1-6)

1

∫ T dt − C !

(6.1-7)

or

S+C = 2

∫

T dt =

∫

m v2 2 dt = 2

∫

! ! m v • v dt !

(6.1-8)

We can write: !

S+C =

Hamilton: !

(6.1-5)

then becomes:

where E is the total energy of the system. !

E dt !

t1

where C is a constant for the given time interval. The action

!

L = 2T − E !

∫

t1

potential energy of the system. Letting:

E = T +V !

2T dt −

t1

t2

the total energy is conserved, and so E remains constant. We

!

!

∫

t2

Since we are considering a stationary conservative force field, then have:

L = T − V = 2T − (T + V ) !

L dt =

t1

6.1! PHYSICAL MEANING OF ACTION

!

∫

t2

∫

! ! dr m v • dt = dt

∫

! ! m v • dr !

(6.1-9)

We then have along the system trajectory: (6.1-4) !

S=

∫ mv ds − C !

(6.1-10)

101

As can be seen from equation (1.2-2), this is essentially the

!

definition of quantity of action proposed by Maupertuis.

elastic system in a stationary conservative force field can only

Finding the stationary value of the integral in equation (6.1-10)

occur when the potential energy of the system is at a minimum.

yields a path that is the same as that obtained from Hamilton’s

Only then will a displacement of the system create a force that

principle. Hamilton’s action is simply the integral of the

will cause the system to return to its original position.

momentum of the system over its path minus a constant:

!

!

S=

∫ L dt = ∫ mv ds − C !

It is not surprising that stable equilibrium of a perfectly

For the principle of least action to be valid for a physical

system requires that the following statements be true: (6.1-11) 1.! Aether exists as a ubiquitous field in our Universe (see Chapters 3 and 4).

6.2! BASIS OF THE PRINCIPLE OF LEAST ACTION

2.! Newton’s laws of motion are valid (see Section 4.4).

!

3.! Euler-Lagrange equations are valid (see Section 5.1.1).

We have seen that the Euler-Lagrange equations can be

obtained from Newton’s laws of motion for a conservative force

4.! Hamilton’s principle results in a minimum (see Sections 2.3.2 and 2.6).

field. Newton’s laws, in turn, are based upon aether field theory. Using the Euler-Lagrange equations, a stationary value for the integral in equation (6.1-10) can be determined. !

Since aether is perfectly elastic, the principle of minimum

total potential energy for perfectly elastic systems subject to conservative forces pertains. From this principle we know that

Therefore the physical foundation of the principle of least action for any system is the aether that is ubiquitous in our Universe. !

Lagrange (1811) was correct when he stated:

when aether is in equilibrium, the potential energy of the aether

“Tel

est

le

principe

auquel

je

donne

ici,

will be a minimum. Therefore the stationary value of the

quoiqu’improprement, le nom de moindre action, et que

integral in equation (6.1-10) must be a minimum. This then

je regarde non comme un principe métaphysique, mais

provides a physical basis for Hamilton’s principle. 102

comme un résultat simple et général des lois de la

today, we must remember that we have access to significantly

mécanique.”

more scientific data now than was ever available in the past.

“Such is the principle to which I give here, albeit improperly, the name least action, and which I regard not as a metaphysical principle, but as a simple and general result of the laws of mechanics.” Clearly Newton’s laws of motion are more fundamental than Hamilton’s principle.

6.3! SUMMARY !

Hamilton’s principle and the principle of least action of

Maupertuis are essentially equivalent for a system within a stationary conservative force field. The physical foundation of the principle of least action is the aether that is ubiquitous in our Universe. Physical laws can be expressed as optimizing principles only because of the nature of the aether. Unification of all the laws of physics is provided by the aether (see Massé, 2022). !

For many years the foundation of the principle of least

action has been a mystery. For much of that time teleological arguments were used to explain the principle (see Osler, 2001). While such teleological thinking may seem nonsensical to us

103

A.1!

Appendix A

!

SOURCE/SINK POINTS AND FIELD POINTS

A source/sink point for a vector field is a point at which is

located a source or a sink of the vector field (see Figure A-1). A field point is any point in the vector field at which a source or a sink of the field does not exist. The field that exists at a field point, which may be located some distance from the source/ sink point, is a direct result of the existence of the source or sink at the source/sink point.

VECTOR FIELD OPERATIONS

!

Vector operations at the field point will now be calculated

using two different coordinate systems: One coordinate system will have its origin at the field point, and the other coordinate system will have its origin at the source/sink point. The

“One geometry cannot be more true than another; it can only be more convenient.” Henri Poincaré Science and Hypothesis

!

In this appendix, we will derive some vector relations that

are useful for the development of certain equations in this book. We will do this by considering vector fields in terms of source/ sink points and field points. We will also develop some relations for the Dirac delta function.

subscript on a term will indicate which of the coordinate systems the term is with respect to. For example, gradient operations at the field point with respect to the field ! coordinates will be written using the operator ∇f , while gradient operations at the field point with respect to the ! source/sink coordinates will be written using the operator ∇s . ! ! If r is the position vector of the field point with respect to the source/sink coordinates, at the field point we then have: ! ! ! ∇f r = ∇s r ! (A.1-1) 104

since the gradient of r is independent of coordinate system. For a scalar field ϕ , we can then use equation (A.1-1) and equation (C-12) of Appendix C to write: !

! ! ∂ϕ ! ∂ϕ ! ∇f ϕ = ∇f r = ∇s r = ∇s ϕ ! ∂r ∂r

(A.1-2)

Equation (A.1-2) then gives: ! ! ! ! ! ! dA ! dA ! ! • ∇f ϕ = • ∇s ϕ = ∇s • A ! ∇f • A = dϕ dϕ From equations (A.1-4) and (A.1-2) we can write: ! ! ! ! ! ! ! ∇f • ∇f ϕ = ∇s • ∇f ϕ = ∇s • ∇s ϕ ! or ! !

!2 !2 ∇f ϕ = ∇s ϕ !

(A.1-4)

(A.1-5)

(A.1-6)

Using equation (A.1-4), we will now write equation (3.4-7)

from Section 3.4.2 as: !

! ! ! ∇s • g = − G ρ ∇f •

∫∫∫

ΔVs

rˆ dVs ! r2

(A.1-7)

or

! ! ∇s • g = − G ρ

Figure A-1! Field point and source/sink point. The field ! position vector r is with respect to source or sink coordinates. !

! If A (ϕ ) is a vector field that is a function of the scalar field

! rˆ ! ∇f • 2 dVs ! (A.1-8) r ΔVs ! Equation (A.1-8) is valid since ∇f operates on field coordinates,

∫∫∫

and so this operator commutes with integration over all source/sink coordinates. We now can use equation (A.1-4) to

ϕ , using the vector relation given in equation (C-25) of

rewrite equation (A.1-8) as:

Appendix C we have: ! ! ! dA ! ! ∇•A = • ∇ϕ ! dϕ

! (A.1-3)

! ! ∇s • g = − G ρ

! rˆ ∇s • 2 dVs ! r ΔVs

∫∫∫

(A.1-9)

or 105

! ! ∇•g = − G ρ

!

! rˆ ∇ • 2 dV ! r ΔV

∫∫∫

(A.1-10)

which is equation (3.4-7).

A.2! ! !

DIRAC DELTA FUNCTION

!

We will now examine the integral:

∫∫∫

⎡1 ⎤ ∇ 2 ⎢ ⎥ dV ! ⎣r ⎦ V

(A.2-1)

(A.2-5)

( r = 0 somewhere in V )! (A.2-6)

and !

∫∫∫ ϕ (r ) δ (r − r ) dV = ϕ (r ) ! 0

0

V

(A.2-7)

we then have from equations (A.2-3), (A.2-5), and (A.2-6):

! ! ⎡1 ⎤ ∇ ⎢ ⎥ = ∇•∇⎢ ⎥ = 0! ⎣r ⎦ ⎣r ⎦ 2 ⎡1 ⎤

(A.2-2)

theorem given in equation (C-58) of Appendix C to obtain:

! ! ⎡1 ⎤ ∇ • ∇ ⎢ ⎥ dV = ⎣r ⎦ V

∫∫∫

! ⎡1 ⎤ ! ∇ ⎢ ⎥ • dS ! S ⎣r ⎦

∫∫

(A.2-3)

where the surface S is taken to be a small sphere surrounding the point r = 0 . Integrating over the sphere and using the gradient operator for spherical coordinates:

! ⎡1 ⎤ ∇ ⎢ ⎥ • eˆr dS = − S ⎣r ⎦

∫∫

∫∫∫ δ (r ) dV = 1! V

To evaluate equation (A.2-1) when r = 0 , we use Gauss’s

!

∫∫

following characteristics:

Appendix C:

!

! ⎡1 ⎤ ∇ ⎢ ⎥ • eˆr dS = − 4 π ! S ⎣r ⎦

If we now define the Dirac delta function δ to have the

If r ≠ 0 , we have the vector relation given in equation (C-51) of

!

!

2π

∫ ∫ 0

π

0

1 2 r sin θ dθ dφ ! r2

!

⎡1 ⎤ ∇ 2 ⎢ ⎥ = − 4 π δ (r ) ! ⎣r ⎦

(A.2-8)

Using equation (C-11) of Appendix C, we can also write: ! ! ⎡ 1 ⎤ ! ⎡ r! ⎤ ! ⎡ rˆ ⎤ 2 ⎡1 ⎤ ! ∇ ⎢ ⎥ = ∇ • ∇ ⎢ ⎥ = ∇ • ⎢− 3 ⎥ = ∇ • ⎢− 2 ⎥ ! (A.2-9) ⎣r ⎦ ⎣r ⎦ ⎣ r ⎦ ⎣ r ⎦ and so from equations (A.2-9) and (A.2-8): ! ⎡ r! ⎤ ! ⎡ rˆ ⎤ ! ∇ • ⎢ 3 ⎥ = ∇ • ⎢ 2 ⎥ = 4 π δ (r ) ! ⎣r ⎦ ⎣r ⎦

(A.2-10)

This equation is used to obtain equation (3.4-8) in Section 3.4.2. (A.2-4)

or 106

Appendix B

The Greek Alphabet

Theta!

θ!

Θ

Iota!

ι!

Ι

Kappa!

κ!

Κ

Lambda!

λ!

Λ

Mu!

µ!

Μ

Nu!

ν!

Ν

Xi!

ξ!

Ξ

Omicron!

ο!

Ο

Pi!

π!

Π

Rho!

ρ!

Ρ

Sigma!

σ!

Σ

Tau!

τ!

Τ

Upsilon!

υ!

ϒ

Alpha!

α!

Α

Phi!

φ, ϕ !

Φ

Beta!

β!

Β

Chi!

χ!

Χ

Gamma!

γ!

Γ

Psi!

ψ!

Ψ

Delta!

δ!

Δ

Omega!

ω!

Ω

Epsilon!

ε!

Ε

Zeta!

ζ!

Ζ

Eta!

η!

Η 107

Appendix C

VECTOR IDENTITIES !

!

! ! ! ! ! ! ! A × (B + C) = A × B + A × C !

(C-2)

! !

! ! ! ! ! ! ! ! ! ! ! ! A• B × C = A × B•C = B•C × A = B × C • A ! ! ! ! ! ! = C • A × B = C × A• B!

(C-3)

!

! ! ! ! ! ! ! ! ! A × ( B × C ) = ( A • C ) B − ( A • B)C !

(C-4)

!

! ! ! ! ! ! A × ( B × C ) ≠ ( A × B) × C !

(C-5)

!

( A × B) × C = ( A • C ) B − ( B • C ) A !

(C-6)

!

! ! ! ! ! ! ! ! ! ! A × ( B × C ) + B × (C × A ) + C × ( A × B ) = 0 !

(C-7)

!

df ! = ∇f • nˆ ! dn

(C-8)

!

! ! ! ∇ ( f g ) = f ∇g + g ∇f !

!

! ! ! ∇r m = m r m−1 ∇r = m r m−1 rˆ = m r m−2 r !

!

! ! ⎛ 1⎞ r rˆ ∇⎜ ⎟ = − 3 = − 2 ! ⎝ r⎠ r r

(C-11)

!

! ! ∂f ( r ) ! ∂f ( r ) r ∂f ( r ) ∇f ( r ) = ∇r = = rˆ ! ∂r ∂r r ∂r

(C-12)

In this appendix are listed some vector identities and

relations that are useful for physics. The following notation is ! ! ! used: A , B , and C are arbitrary vectors; ax , ay , and az are ! rectangular components of the A vector; f and g are arbitrary ! scalar functions; and m is an integer. The vector r is the position vector. The vectors iˆ , ˆj , and kˆ are unit vectors along the x, y, and z rectangular coordinate axes, respectively, and nˆ is a unit normal vector.

! ! A × B = ay bz − az by iˆ + ( az bx − ax bz ) ˆj + ax by − ay bx kˆ ! (C-1)

(

)

(

!

!

!

!

! !

!

! !

(C-9) (C-10)

)

108

!

! ∂g ! ∇g ( f ) = ∇f ! ∂f

(C-13)

!

!

! ! ∂bx ∂by ∂bz ! ∇• B = + + ∂x ∂y ∂z

(C-14)

!

!

! ! ! ! ! ! ! ∇ • ( A + B) = ∇ • A + ∇ • B !

(C-15)

!

! ! ! ! ! ! ∇ • ( f A ) = A • ∇f + f ∇ • A !

!

! ! ! ! 2 ∇ • f ∇g = f ∇ g + ∇f • ∇g !

!

! ! ∇ • r = 3!

!

! 2 ∇ • rˆ = ! r

!

! ! ∇ • r m r = ( m + 3) r m !

! ! ! !

(

)

(

)

(

! ! dr • ∇ f = df !

(

! ! ! ! A • ∇ f = A • ∇f !

)

)

(

! ! ! ! ! ! ∂B ∂B ∂B ! A • ∇ B = ax + ay + az ∂x ∂y ∂z

(

! ! ! ! A•∇ r = A!

) )

! ! ! dA ! ∇ • A( f ) = • ∇f ! df

(C-25)

! ! ⎡ ∂b ∂by ⎤ ˆ + ⎡ ∂bx − ∂bz ⎤ ˆj + ⎡ ∂by − ∂bx ⎤ kˆ ! (C-26) ∇× B= ⎢ z − i ⎥ ⎢ ∂x ∂y ⎥ ⎢⎣ ∂z ∂x ⎥⎦ ⎣ ∂y ∂z ⎦ ⎣ ⎦

!

! ! ! ! ! ! ! ∇ × ( A + B) = ∇ × A + ∇ × B !

(C-27)

(C-16)

!

! ! ! ! ! ! ∇ × ( f A ) = ∇f × A + f ∇ × A !

(C-28)

(C-17)

!

! ! ! ∇ × ∇f = 0 !

(C-29)

(C-18)

!

! ! ! ! ∇ × f ∇g = ∇f × ∇g !

(C-30)

(C-19)

!

! ! ! ∇ × f ∇f = 0 !

(C-31)

!

! ! ! ∇ • ∇f × ∇g = 0 !

(C-32)

!

! ! ! ∇×r = 0!

(C-33)

!

! ! ! ∇ × ( f (r ) r ) = 0 !

(C-34)

(C-20) (C-21) (C-22)

)

(

)

(

)

! ! ! ! A × ∇ f = A × ∇f !

!

(

!

! ! ! ! dA ! ∇ × A ( f ) = ∇f × df

(C-23) (C-24)

(

)

(C-35) (C-36)

109

!

! ! ! ∇•∇ × A = 0!

!

! ! ! ! ! ! ! ! ! ∇ • ( A × B) = B • ∇ × A − A • ∇ × B !

(C-37)

(

(

!

)

(

)

(

)

(C-39)

! ! ! ! ! ! ∇ • f ∇ × A = ∇ × A • ∇f !

(

!

) (

)

) (

) (

) (

)

!

! ! ! ! ! ! ! ! ! ∇ ( A • B) = A • ∇ B + B • ∇ A ! ! ! ! ! ! +A × ∇ × B + B × ∇ × A !

!

! ! ! ! ! ! ! ! ! ! ∇ × ∇ × A = ∇ × ∇ × A = ∇ ∇ • A − ∇ 2A !

(

!

! !

(

) (

)

(

)

)

(

(

)

)

(C-42) (C-43)

! ! ! ! ! ! ! ! ! ! ! ! ∇ • A × ( B × C ) = ( A • C ) ∇ • B − ( A • B) ∇ • C ! ! ! ! ! ! ! ! + B • ∇ ( A • C ) − C • ∇ ( A • B ) ! (C-44)

(

)

(

)

(

)

!

(

! ! ! ! ! ! 1! 2 ∇ × A × A = A • ∇ A − ∇A ! 2

(C-45)

!

! ! ! 1! ! ! ! A × ∇ × A = ∇A 2 − A • ∇ A ! 2

(C-46)

!

)

(

(

(

)

)

(

)

! ! ! 1! ! ! ! A × ∇ × A = ∇A2 − A ∇• A ! 2

)

(

! ! ∇ • ∇f = ∇ 2 f !

(C-49)

!

∇ 2r m = m ( m + 1) r m−2 !

(C-50)

!

⎛ 1⎞ ∇2⎜ ⎟ = 0! ⎝ r⎠

(C-51)

!

∇ 2r 2 = 6 !

(C-52)

!

! ! ! ! ! ! ! ∇ A = ∇ ∇• A −∇ × ∇ × A !

(C-53)

!

! ! ∇ 2 ∇f = ∇ ∇ 2 f !

(C-54)

!

! ! ! ! ∇ • ∇ 2A = ∇ 2 ∇ • A !

(C-55)

!

! ! ! ! ! ! 2 ∇ (A • r ) = 2∇ • A + r • ∇ A!

(C-56)

!

! ! ∇ ( f g ) = g∇ f + 2 ∇f • ∇g + f ∇ 2g !

(C-57)

(C-40)

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ∇ × ( A × B ) = ∇ • B A − ∇ • A B + B • ∇ A − A • ∇ B ! (C-41)

(

!

(C-38)

! ! ! ! ! ! ! ! ! ∇ × A • B = ∇ • ( A × B) + A • ∇ × B !

)

!

∂2 f ∂2 f ∂2 f ∇ f= 2+ 2 + 2! ∂x ∂y ∂z

)

(

2

(

( )

(

)

(

(C-48)

)

)

)

(

)

2

2

2

Divergence theorem (Gauss’s theorem): !

(C-47)

2

∫∫∫

! ! ∇ • B dV =

V

∫∫

! ! B • dS !

(C-58)

S

110

!

! ! ∇ • B dV =

∫∫∫

! B • nˆ dS !

∫∫

V

S

(C-59)

Green’s first theorem: !

! ! ∇ × B dV =

∫∫∫

V

!

∫∫

! ! ∇ × B dV = −

∫∫∫

! nˆ × B dS !

(C-60)

∫∫

! ! B × dS !

S

! (C-61)

V

∫∫

! f dS !

(C-62)

S

Stokes’s theorem:

∫∫

! ! ! ∇ × B • dS =

S

!

−

"∫

! ! B • dr !

"∫

! f dr !

C

∫∫

S

! ! ∇f × dS =

(C-65)

S

∫∫∫ ( f ∇ g − g∇ f ) dV = ∫∫ ( 2

2

S

df ⎤ ⎡ dg f − g dS ! ⎢⎣ dn ⎥ dn ⎦

(C-66)

! ! f ∇g − g ∇f • nˆ dS ! (C-67)

)

Reynolds’ transport theorem: !

!

∫∫∫

2

V

Gradient theorem:

∫∫∫

S

( f ∇ g − g∇ f ) dV = ∫∫ 2

V

!

! ∇f dV =

∫∫

! ! f ∇g • dS !

Green’s second theorem:

S

V

!

)

V

Curl theorem: !

∫∫∫ (

! ! f ∇ 2g + ∇f • ∇g dV =

D Dt

∫∫∫

ρα dV =

V

∫∫∫

V

ρ

Dα dV ! Dt

(C-68)

where D Dt is the substantive derivative, ρ is the (C-63)

density of the medium having no sink or source in volume V , and α is some property of the medium

(C-64)

(see Appendix F).

C

111

f g )′ dt = ( f ′ g + f g′ ) dt ! ( ∫ ∫

!

Appendix D

(D-2)

or

fg=

!

∫ f ′ g dt + ∫ f g′ dt !

(D-3)

The constant resulting from integration of the left side of this equation is omitted since it will be combined with the constants from the remaining integrals. Rewriting equation (D-3), we have:

Integration by Parts

!

∫ f g′ dt = f g − ∫ f ′ g dt !

(D-4)

Now we can let: !

In this Appendix we describe the process of changing a

function that we wish to integrate into another function that

!

!

!

Let f ( x ) and g ( x ) be two functions of x , and let both

functions have first-order derivatives. We can the write: !

( f g )′ = f ′ g + f g′ !

Integrating this equation, we have:

(D-1)

dv = g′ ( t ) dt !

(D-5)

We then have:

can be easier to integrate. This process is known as integration by parts.

u = f (t ) !

du = f ′ ( t ) dt !

v=

∫ g′ (t ) dt = g (t ) !

(D-6)

Equation (D-4) becomes: ! !

∫ u dv = u v − ∫ v du !

(D-7)

This is the equation resulting from integration by parts.

For any particular equation it is necessary to select u and dv . 112

After dv is integrated to obtain v , equation (D-7) can be used to obtain the solution of the original integral. Note that the constant of integration in equation (D-6) is taken to be zero since, if it is nonzero, it will be incorporated in the constant of integration of the integral in equation (D-7). Example D-1 Evaluate the integral: !

L

{ f (t )} = Tlim →∞ ∫

T

0−

e− s t f ( t ) dt

Solution: Let: ! !

u = f (t ) !

dv = e− s t dt

du = f ′ ( t ) dt !

e− st v=− s

We then have: !

lim

T →∞

∫

T

∫

T

0−

e

−st

⎡ 1 f ( t ) dt = lim ⎢ − f ( t ) e− s t T →∞ ⎢ s ⎣

and so: !

lim

T →∞

0−

e

−st

f ( t ) dt =

( )+1

f 0− s

s

∫

∞

0−

∞

1 + s 0−

⎤ e− s t f ′ ( t ) dt ⎥ 0− ⎥⎦

∫

T

e− s t f ′ ( t ) dt 113

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