Notes on Dynamical Systems 9780821835777, 9781470431129

Table of contents :
Preface
Chapter 1. Transformation Theory
1.1. Differential Equations and Vector Fields
1.2. Variational Principles, Hamiltonian Systems
1.3. Canonical Transformations
1.4. Hamilton-Jacobi Equations
1.5. Integrals and Group Actions
1.6. The SO(4) Symmetry of the Kepler Problem
1.7. Symplectic Manifolds
1.8. Hamiltonian Vector Fields on Symplectic Manifolds
Chapter 2. Periodic Orbits
2.1. Poincare's Perturbation Theory of Periodic Orbits
2.2. A Theorem by Lyapunov
2.3. A Theorem by E. Hopf
2.4. The Restricted 3-Body Problem
2.5. Reversible Systems
2.6. The Plane 3- and 4-Body Problems
2.7. Poincare-Birkhoff Fixed Point Theorem
2.8. Variations on the Fixed Point Theorems
2.9. The Billiard Ball Problem
2.10. A Theorem by Jacobowitz and Hartman
2.11. Closed Geodesies on a Riemannian Manifold
2.12. Periodic Orbits on a Convex Energy Surface
2.13. Periodic Orbits Having Prescribed Periods
Chapter 3. Integrable Hamiltonian Systems
3.1. A Theorem of Arnold and Jost
3.2. Delaunay Variables
3.3. Integrals via Asymptotics; the Stormer Problem
3.4. The Toda Lattice
3.5. Separation of Variables
3.6. Constrained Vector Fields
3.7. Isospectral Deformations
Bibliography

Citation preview

Notes on Dynamical System s

Couran t Lectur e Notes in Mathematic s Executive Editor Jalal Shatah Managing Editor Paul D. Monsour Assistant Editor Reeva Goldsmith Suzan Toma Copy Editor Josh Singer

http://dx.doi.org/10.1090/cln/012

Jtirgen Moser Eduard J. Zehnder ETH Zurich

12 Note

s on Dynamical Systems

Courant Institute of Mathematical Science s New York University New York, New York American Mathematical Societ y Providence, Rhode Island

200 0 Mathematics Subject

Classification.

Primar y 37-01 , 37Kxx , 53Dxx , 58Exx , 70Fxx , 70H05 .

Fo r additiona l informatio n an d update www.ams.org/bookpages/cln-1

s o n thi s book , visi t 2

Librar y o f Congres s Cataloging-in-Publicatio n Dat a Moser , Jiirgen , 1928 Note s o n dynamica l system s / Jiirge n Moser , Eduar d J . Zehnder . p. cm . — (Couran t lectur e note s i n mathematic s ; 12 ) Include s bibliographica l references . ISB N 0-8218-3577- 7 (alk . paper ) 1. Differentiabl e dynamica l systems . 2 . Hamiltonia n systems . 3 . Transformation s (Mathe matics ) 4 . Combinatoria l dynamics . I . Zehnder , Eduard , 1940 - II . Title . III . Series . QA614.8.M6 7 200 5 515'.39— dc22 200505587

1

Copyin g an d reprinting . Individua l reader s o f thi s publication , an d nonprofi t librarie s actin g fo r them , ar e permitte d t o mak e fai r us e o f th e material , suc h a s t o cop y a chapte r fo r us e in teachin g o r research . Permissio n i s grante d t o quot e brie f passage s fro m thi s publicatio n i n reviews , provide d th e customar y acknowledgmen t o f th e sourc e i s given . Republication , systemati c copying , o r multipl e reproductio n o f an y materia l i n thi s publicatio n is permitte d onl y unde r licens e fro m th e America n Mathematica l Society . Request s fo r suc h permissio n shoul d b e addresse d t o th e Acquisition s Department , America n Mathematica l Society , 201 Charle s Street , Providence , Rhod e Islan d 02904-2294 , USA . Request s ca n als o b e mad e b y e-mai l t o [email protected] . © 200 5 b y th e authors . Al l right s reserved . Printe d i n th e Unite d State s o f America . © Th e pape r use d i n thi s boo k i s acid-fre e an d fall s withi n th e guideline s establishe d t o ensur e permanenc e an d durability . Visi t th e AM S hom e pag e a t http://www.ams.org / 10 9 8 7 6 5 4 3 2 1 1

0 09 08 0 7 06 0 5

Contents Preface vi

i

Chapter 1. Transformatio n Theory 1 1.1. Differentia l Equations and Vector Fields 1 1.2. Variationa l Principles, Hamiltonian Systems 1 1.3. Canonica l Transformations 1 1.4. Hamilton-Jacob i Equations 2 1.5. Integral s and Group Actions 4 1.6. Th e SO(4) Symmetry of the Kepler Problem 5 1.7. Symplecti c Manifolds 6 1.8. Hamiltonia n Vector Fields on Symplectic Manifolds 7

1 9 9 1 2 2 3

Chapter 2. Periodi c Orbits 8 2.1. Poincare' s Perturbation Theory of Periodic Orbits 8 2.2. A Theorem by Lyapunov 9 2.3. A Theorem by E. Hopf 9 2.4. Th e Restricted 3-Body Problem 10 2.5. Reversibl e Systems 10 2.6. Th e Plane 3- and 4-Body Problems 11 2.7. Poincare-Birkhof f Fixe d Point Theorem 12 2.8. Variation s on the Fixed Point Theorems 13 2.9. Th e Billiard Ball Problem 14 2.10. A Theorem by Jacobowitz and Hartman 14 2.11. Close d Geodesies on a Riemannian Manifold 16 2.12. Periodi c Orbits on a Convex Energy Surface 17 2.13. Periodi c Orbits Having Prescribed Periods 18

5 5 4 8 2 8 6 0 3 0 9 0 2 1

Chapter 3. Integrabl e Hamiltonian Systems 18 3.1. A Theorem of Arnold and Jost 18 3.2. Delauna y Variables 19 3.3. Integral s via Asymptotics; the Stormer Problem 20 3.4. Th e Toda Lattice 21 3.5. Separatio n of Variables 22 3.6. Constraine d Vector Fields 23 3.7. Isospectra l Deformations 24

7 7 9 7 3 8 4 2

Bibliography 25

5

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Preface Written i n 1979-80 , thes e notes constitute th e first three chapters o f a book that was never finished. It was planned as an introduction to the field of dynamical systems, i n particular, o f the special class of Hamiltonian systems . W e aimed a t keeping the requirements of mathematical techniques minimal but giving detailed proofs and many examples and illustrations from physics and celestial mechanics. After all, the celestial iV-body problem is the origin of dynamical systems and gave rise in the past to many mathematical developments. The first chapter is about the transformation theory of systems and also contains the so-called Hamiltonian formalism. The second chapter is devoted to periodic phenomena and starts with perturbation methods going back to H. Poincare and local existence results due to Lyapunov and E. Hopf. Classical periodic solutions are established in the restricted 3-body problem and the celestial 3- and 4-body problems. Variational techniques then allow searching for global periodic orbits lik e closed geodesies o n Riemannian manifolds an d closed orbits o n convex energy surface s of general Hamiltonian systems. Th e Poincare-Birkhoff fixed point theorem of an area-preserving annulus map in the plane is also proven in the second chapter. This theorem led to the V. Arnold conjectures about forced oscillations of time-periodic Hamiltonian systems on symplectic manifolds. Incidentally, after these notes were written, th e Arnold conjectures triggere d new developments i n symplectic geom etry an d Hamiltonian systems . Also , i t turned ou t tha t th e periodi c phenomen a of Hamiltonian systems are intimately related to symplectic invariants and surprising symplectic rigidity phenomena discovered b y Y. Eliashberg an d M . Gromov. These more recent developments ar e presented i n the book Symplectic Invariants and Hamiltonian Dynamics by H. Hofer and E. Zehnder. The third chapter is devoted t o a special and interesting class of Hamiltonian systems possessing many integrals. Following the construction of the so-called action and angle variables, illustrated by the Delaunay variables, several examples of integrable systems are described in detail. Chapters 4 and 5 should have dealt with the analytically subtle stability problems in Hamiltonian systems close to integrable systems known as KAM theory, and with unstable hyperbolic solutions, which, in general, d o coexist with th e stable solutions. Unfortunately , thes e chapters wer e never completed. These notes owe much to Jiirgen Moser's deep insight into dynamical systems and his broad view of mathematics. They also reflect his specific approach to mathematics by singling out inspiring typical phenomena rather than designing abstract theories. Finally, I would like to thank Paul Wright for carefully checking these notes. vii

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http://dx.doi.org/10.1090/cln/012/01

CHAPTER 1

Transformation Theor y 1.1. Differential Equations and Veetor Fields (a) The flow of a system of differential equations. Th e object of these lecture notes are systems of ordinary differential equations of the form dx (i.i ) :T 7= / M at or in components, dx' = fj(x) > • ' = 1 '•••*' , ' (1.10 ~di defined i n an open domain Del" . Th e right-hand sid e f(x) i s a vector-valued function mapping D into W1, belonging to C r(D,Rn) fo r r > 1 . We recall the wellknown fact which will not be proven here that system (1.1) has a unique solution x(t) for a given initial value x(0) e D, where the solution x(t) i s defined o n an interval \t| < 5 , 8 > 0. Mor e precisely, if K i s a compact subset of D , then there exists a 8 > 0 depending on K and / suc h that the solution x(t) wit h initial values x(0) e K exists for the interval I = {t:\t\-z

t=0

= [X,Z].

Hint: Differentiate y lh(

0

is th e semimajor axi s o f the ellipse an d e ( 0 < e < 1) is the eccentricity o f the ellipse. Th e angle determines the position of the ellipse. Moreover , th e distance r = (x2 + y2)1/2 i s given by r = a(l -j-ecosu). 1.2. Variational Principles, Hamiltonian Systems In many fields of applications, the differential equations are derived from a variational principle and consequently have special features whic h we will investigate in this section. (a) Variational principles . W e begin wit h th e standard variational principl e and its Euler equation. W e consider a function F(t,q,r) o f 2n + 1 variables t e E , q,r € Kw, defined i n a domain Q = I x K 2 \ / = {t : t\ < t < t 2}, and form the functional

(1.13) [q]

= J2F(t,q(t),q(Wt, q

For example, A. Wintner [47], with some change of notation.

=^

12 1

. TRANSFORMATION THEORY

for curve s q e C 2(I,Rn) wit h fixed en d point s q{t\) = a, q(ti) = b. Th e rathe r ugl y choic e of th e letter s q = (q\ , qi,... ,q n) i s dictate d b y tradition ! This functiona l assign s to any curv e q e C 2(I,Rn) a real number . Th e stationar y value s o f thi s functiona l ar e obtaine d b y formin g th e first variatio n »t2

&[q,q]= f\(F q,q) +

{F 4^))dt

of an d requirin g tha t it vanishe s fo r ever y choic e of q e C l(I9Rn). Integratio part s lead s t o th e Eule r equation s (1.14 ) —(Fq(t

9q9q))

=

F

n by

q(t,q,q)

for q = q(t) an d q = j- tq(t), whic h w e wil l investigate . Thi s i s a syste m o f secon d orde r 72 Fq4 qq 2+Fqq

Fq dt ~0, dt^ " dt+ dtqdt+ dt wher e w e omitte d th e argument s o f th e derivative s o f F. On e call s a poin t (t,q,q) el x R 2n "regular " if

(1.15 ) det(F^)^

0

and w e wil l impos e thi s assumptio n o f regularity . I t implie s tha t w e ca n writ e th e Eule r equatio n i n th e explici t for m q — G(t,q,q) and henc e a s a syste m of first orde r

±(q) = ( q

dt \q) \G(t,q,q);

(b) Legendr e transformation . Ther e i s a mor e elegan t wa y t o achiev e thi s re ductio n t o a first-order syste m of differentia l equation s whic h lead s to the Hamilton ian for m o f th e equation . Thi s is achieve d b y th e so-calle d Legendr e transformatio n accordin g t o whic h on e introduce s instea d o f q th e vecto r variabl e (1.16 ) P

= Fq o r Pj

=F

4j(t,q,q).

All th e followin g consideration s ar e "local " i n natur e sinc e w e appea l t o th e implici t functio n theorem . Th e Legendr e transformatio n map s (t9q,q)-+ (t,q,p) wher e p i s defined b y (1.16) . Th e Jacobia n of thi s mappin g i s clearl y de t Fqq; whic h by (1.15 ) i s assume d no t to vanish . Therefor e th e invers e mappin g exist s locally : (t,q,p)-+(t,q,q) wit

hq

= V(t,q,p).

Thi s mappin g ca n b e expresse d effectivel y i f on e introduce s th e Hamilto n functio n H(t9q9p)=(p9V)-F(t,q9V)9 or equivalently , F(t9q,q) =

(Fq,q)-H(t

9q9Fq).

1.2. VARIATIONAL PRINCIPLES, HAMILTONIAN SYSTEM S

13

In other words we have H(t,q,p) = (p,q) - F(t,q,q) if p = Fq o r q = V . Therefore taking the differential, wher e we consider at first q and p independently and then set p = Fq, we find dH = (p,dq) - h {q,dp} - (F q,dq) - (Fq,dq) - F tdt = {p- Fq.dq) + (q,dp) - {F q,dq) - F tdt = (q,dp) - (F q,dq) -F tdt. Thus (1.17) H

p

= q, Hq

= -F q, H

t

= -F t.

The first relation shows that V(t,q9p) = H

p(t,q,p).

These calculations show that the Euler equations are transformed int o the system 0-18) \"

= H

/

This form of the equations is called Hamilton's form of the differential equations , or Hamiltonian systems for short. We have to keep in mind that this derivation was only local in nature—but we will see that in many applications the Legendre transformation i s linear in q an d is meaningful i n the large. Moreover , on e frequently considers the Hamiltonian system as the primary object, rather than the variational principle. (c) Autonomous case . I f we assume F to be independent of t, the same is true for H an d we obtain an autonomous Hamiltonian system, which we will write in the form (1.1) of §1 . Fo r this purpose we introduce the 2n vector x an d the 2n by 2n matrix J by

--(;) • '-(

io

where / = / „ is the n by n identity matrix. Then the Hamiltonian system (1.18) can be written in the form dx (1.19) — = JVH at where VH i s the vector with components H Xj; i.e. , th e gradient o f H. I n other words, WH is defined by the differential relation dH = (VH,dx). Thus Hamiltonian systems are vector fields (1.1) where f = JVH is defined i n terms of a single function, th e Hamiltonian. Thi s suggests that these systems are "special" in nature. This is indeed the case as we shall see.

1. TRANSFORMATION THEORY

14

(d) Canonical transformations. I f we subject system (1.19) to an appropriate change of coordinates, the special form of the equation will be preserved and we will determine a class of transformations whic h preserve the class of Hamiltonian systems. Since J 2 = —hn we can write (1.19) also in the form (1.20) -Jx

= VH.

If we subject this system to an arbitrary transformation x = u(y), K(y)

= H(u(y)),

we find x = uyy, VK

= (uy) VHou,

T

where (-) denotes the transposed matrix. Thus (1.20) goes into —uyJuyy = WK. Hence if we require the identity T yJuy

(1.21) u

= J,

then the transformed system is y = JVK with the Hamiltonian K = H o u obtained from H simply by transformation. Thu s transformations x = u(y) satisfying th e identity (1.21 ) d o preserve th e class of Hamiltonian systems. DEFINITIO N A diffeomorphism x = u(y) is called canonical if it satisfies (1.21). The canonical transformations for m a group which we will investigate in the next section. (e) Examples. EXAMPL E 1 . I f n = 1 an d

H = -p —cosq, then the Hamiltonian system gives rise to the pendulum equation q + sinq = 0. EXAMPL E

2 . Th e equations of particle mechanics are written in Hamiltonian

form j =\ J

where U(q) is called th e potential an d mj > 0 are the masses. Th e differential equations become (1.22) mtfj^-Uq.,

j

= l,2,...,n .

Notice that in this case the Legendre transformation i s pj = m 74j which is globally invertible.

1.2. VARIATIONAL PRINCIPLES, HAMILTONIAN SYSTEMS 1

5

The iV-body problem in E 3 is contained in the above formulation: I f n = 3N, m^j-2 = JW3/-1 = msj = jjij > 0 is the mass of the j t h particl e and >

Qj = Qv-i

\W / its position, we have the potential N

U(q) = -J2 and (1.22) becomes

which are the equation of motion of the iV-body problem. EXAMPL E

3 (Geodesies). I f we introduce the metric n

2

ds = 2_\, 8ij( x)dxidxf where g(x) = {gij(x)) is a positive definite symmetric matrix, the geodesies can be defined as the extremals of the functional / ~(g(x)x,x)dt. The Euler equations d

(

dt

/

1

—

aj J

v ^ dgjk

2^-f

j jM

dx x

*k

can be written as

L^+E(g-^)^=»j j£

Using the obvious identity

J

j,k j,k

we can write the equation in the form ^Sijxj +

J2 TnkXjXk = 0

where r

..= Jl

2

l

(d&J ,

8

^d

Sjk\

\ 3x& 3x y 3x / /

1. TRANSFORMATION THEORY

16

We use the notation of differential geometr y and denote the inverse matrix g 1 by (glJ) and introduce the Christoffel symbol s

so that the Euler equation becomes

We were inconsistent in the notation describing points by the letter x instead of q. Th e Legendre transformation i s given by p = g(x)x which is clearly invertible since detg > 0, and the Hamiltonian is given by H(x,p)=~(g~1(x)p,p). EXAMPL E 4 (Charged particle in an electromagnetic field). We consider a particle of mas s m > 0 and charge e i n a n electromagnetic field, where th e electric potential is q + -qAB

where B(q) = VAA(q) is the magnetic field, and c is the speed of light. 3 Also these equations are the Euler equation of a variational problem with the Lagrange function, (1.23) F(q,q)=^\q\

2

-e + ~(q,A). 2c Therefore they can also be written in Hamiltonian form if p'= Fq =mq + -A c is introduced. Again the Legendre transformation i s linear in q and globally invertible. One computes

p--A(q)

+# . 2m c As a special case we describe the equation of a charged particle in a magnetic dipole field (like that of the earth). In this case the vector potential is A = CAV(\q\~

l

)

where C has the direction of the dipole axis and \C\ measures its strength. 3

V A A or curl A is the vector with components Mi-M2,Mi-|li,and|^-

-

1.2. VARIATIONAL PRINCIPLES, HAMILTONIAN SYSTEMS

17

If C = e 3 one verifies that B = VAA = -~ — V(\qr l) = V(q 3\qr3) dq3 so that the differential equations are e( q mq = ~qAV[ — c \\qY The Hamiltonian is in this case 1 p-'-A H= 2m c 1 + p\ \ wher e r c r3 cr 3 / I 2m EXAMPL E 5 (The relativistic equations). Actually the above equations are valid only for velocities \q\ small compared to the speed of light c. The relativistic equations are described by the Lagrange function

F(q9q) = mc 2(lx 7- jl-c~ 2\q\2) -e4> + -(q,A) . c If one expands this function i n powers of c""1 \q\ and drops terms of at least fourth order, on e obtains (1.23) . Therefor e th e Euler equation o f thes e tw o variational problems are "close" if c~~lq is small. The variables p is introduced by p = F4 =

mq

yi-c- 2 kj|

2

+ S -A.

c

This relation is not linear in q any more; moreover, it has to be restricted to speed 141 x = u(y) from R k int o Em w e call the resulting 2-form in R k u*a(y) = (u ydy9A(u(y))(AUydy)) — [dy,u

y(Aou)uy(Ady))

= (dy,B(y)(Ady))

9

so that u*a is represented by the matrix (1.32) B(y)

= u

T y(Aou)uy.

More conceptually the 2-form w*a at y is the antisymmetric bilinear form on R*, u*a(y)(v, w) = a(x)(Uv, Uw) , x = u(y) 9 where U = u y, an d v,w e R k. I n particular, i f u is a coordinate trans formation in E m with u y nonsingular, we have, recalling the transformation law for vector fields /(#)> g(x) o n Rm> u*a(y)(ji*f(y)9u*g(y)) = «(*)(/(*),*(*))

.

x = w(y), expressing the coordinate independence. We apply these remarks to the special 2-form on R2w, &) = - (J dx,Adx) = - /^(dpj A dqj — (t;,u; ) = {Jv,w) fo r u,u ; € M2w. This leads to the equivalent DEFINITIO N

A mapping x = u(y) i s canonical precisely if

(1.33) u*Q)

= co

where n

(1.34) &

> = Y^ dpj A dqj . j=i

This differential form , called the symplectic form, takes the place of the antisymmetric nondegenerate bilinear form i n the linear case . I t is a closed 2-form; i.e. , dco = 0 , which i s nondegenerate. W e will se e that every 2-form wit h thes e tw o properties can be transformed by appropriate local coordinates into the form (1.34). But in this section we shall continue to work with the differential form (1.34). (c) Poisson brackets. Ther e is a third and most important way to characterize canonical mappings. For any two smooth functions F, G we define the function {F9G} = (F

p,Gq)-(Fq9Gp)

where again Xj = qj, Xj +n = pj for j = 1,2,.. . ,n, or {F,G} = (JVF

iVG}.

24 1

. TRANSFORMATION THEORY

LEMMA 1.5 A mapping x = u(y) is canonical if and only if the Poisson brackets transform like {FyG}ou = {F ou.Gou) for all functions F,G. PROOF :

Sinc e V(Fou) = u

T yVFou

we find {Fou,Gou) =

(Jw VF

OU, U VGOM

)

= (u yJu VF

ou.VGou )

and the condition of the lemma requires that ul b e a symplectic matrix. But we saw that U is symplectic if and only if U T is , proving the lemma. • LEMMA 1.6 The Poisson bracket satisfies the identities {{F,G},#

{F,G} = ^{G,F} and

} + {{G,#},F } + {{/f,F},G } = 0

for all smooth functions F , G, H. The first relation i s obvious, the second, called the Jacobi identity, requires a calculation which we leave to the reader. With the notation of Section 1.1 we will associate with any function F = F(x ) a Hamiltonian system x = JVF or the partial differential operator

XF=

hvpjH^Fqj^) 7=1

which also represents our vector field. With this notation XFG = (F PjGq) - {F q,Gp) = +{F,G}. Hence we have (1.35) X

GF

= {G,F} = -X

FG.

It is easy to see now that the Hamiltonian vector fields form a Lie algebra, i.e., that the commutator [Xf, XGI =

X FXG

— XgXf

is again a Hamiltonian vector field. In fact, (1.36) [X

F,XG]

= X H wit h H

= {F,G).

To verify thi s apply both sides to a function, sa y h, an d apply (1.33) and the Jacobi identity.

1.3. CANONICAL TRANSFORMATIONS

25

(d) Flow of a Hamiltonian vector field. THEORE M

1.7 A vector field

* = £/>(*) =i 9

3 ^'

is Hamiltonian if and only if the corresponding flow