Convexity Methods in Hamiltonian Mechanics [Paperback ed.] 3642743331, 9783642743337

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Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge . Band 19 A Series of Modem Surveys in Mathematics

Editorial Board

E. Bombieri, Princeton S. Feferman, Stanford N.H.Kuiper, Bures-sur-Yvette P. Lax, New York H. W. Lenstra, Jr., Berkeley R Remmert (Managing Editor), Munster W. Schmid, Cambridge, Mass. J-P. Serre, Paris 1. Tits, Paris K. K. Uhlenbeck, Austin

Ivar Ekeland

Convexity Methods in Hamiltonian Mechanics

Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong

Ivar Ekeland CEREMADE, Universite Paris-Dauphine F-75775 Paris Cedex 16, France

Mathematics Subject Classification (1980): 58Exx, 58Fxx, 70Hxx, 70Jxx, 70Kxx ISBN-13: 978-3-642-74333-7 001: 10.1007/978-3-642-74331-3

e-ISBN-13: 978-3-642-74331-3

Library of Congress Cataloging-in-Publication Data Ekeland, I. (lvar), 1944- Convexity methods in Hamiltonian mechanics I Ivar Ekeland, p. cm.(Ergebnisse der Mathematik und ihrer Grenzgebiete; 3. Folge, Bd. 19) Bibliography: p. Includes index.

ISBN-13: 978-3-642-74333-7 I. Hamiltonian systems. 2. Convex domains. I. Title II. Series.

QA614.83.E44 1990 514'.74-dc20

89-11405

This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law.

© Springer-Verlag Berlin Heidelberg 1990 Softcover reprint of the hardcover 1st edition 1990 2141/3140-543210 - Printed on acid-free paper

Table of Contents

Chapter I. Linear Hamiltonian Systems ...............................

1

Floquet Theory and Stability ..................................... Krein Theory and Strong Stability ................................ Time-Dependence of the Eigenvalues of R(t) ...................... Index Theory for Positive Definite Systems ........................ The Iteration Formula ............................................ The Index of a Periodic Solution to a Nonlinear Hamiltonian System Examples ......................................................... Non-periodic Solutions: The Mean Index..........................

1 7 15 23 34 54 65 74

Chapter II. Convex Hamiltonian Systems .............................

79

1. Fundamentals of Convex Analysis .................................

2. Convex Analysis on Banach Spaces................................ 3. Integral Functionals on LOi. ..•..••...•••••.••••••••••.•...••.•...•. 4. The Clarke Duality Formula ......................................

79 86 93 98

Chapter III. Fixed-Period Problems: The Sublinear Case..............

110

Introduction ...................................................... An Existence Result .............................................. Autonomous Systems............................................. Nonautonomous Systems .......................................... Other Problems ...................................................

110 111 117 121 129 133

Chapter IV. Fixed-Period Problems: The Superlinear Case............

136

Introduction ......................................................

136 136 148 157

1. 2. 3. 4. 5. 6. 7. 8.

1. Sub quadratic Hamiltonians .......................................

2. 3. 4. 5.

1. Mountain-Pass Points.............................................

2. A Preliminary Existence Result ................................... 3. The Index at Mountain-Pass Points...............................

vi

Table of Contents

4. Subharmonics..................................................... 5. Autonomous Problems and Potential Wells ........................

171 175

Chapter V. Fixed-Energy Problems...................................

187

Introduction ...................................................... 2. Multiplicity in the Pinched Case .................................. 3. Multiplicity in the General Case .................................. 4. Open Problems ...................................................

187 187 198 215 235

Bibliography .........................................................

237

Index................................................................

245

1. Existence, Length, Stability .......................................

Introd uction

The study of periodic solutions of Hamiltonian systems has a long and prestigious history. It starts with the invention of calculus, when Isaac Newton wrote down the differential equations derived from the Hamiltonian II!II' and found their solution to be the Keplerian ellipses. The "Principia mathematica philosophiae naturalis", published in 1687, record this discovery and lay the foundations of western science. Two hundred years later appeared Henri Poincare's famous treatise, the "Methodes nouvelles de la mecanique celeste" , which recognized for the first time the inherent complexity of nonintegrable Hamiltonian systems, and opened the way to the modern theory of dynamical systems. In another book l I have tried to describe the intellectual posterities of Newton and of Poincare. Their opposition appears already in their mathematics. Newton dealt with the Kepler problem, all the solutions of which are periodic (provided the energy level is negative). After him, attention focused on completely integrable systems, and on the regularities that could be found in the motion of mechanical systems. After two centuries of work by astronomers on the three-body problem, Poincare proved that a complete description of the motion was impossible, and showed that most Hamiltonian systems would contain trajectories with wildly irregular behaviour. Attention then slowly started shifting to nonintegrable systems and to ergodic properties, culminating in today's interest in chaos. Periodic solutions are relevant to both sides to the story. On the one hand, they are paradigms of regularity, the simplest solutions one can find apart from equilibria. On the other hand, a tubular neighbourhood of a periodic solution may contain examples of all kinds of trajectories, as illustrated by today's computer pictures, which show elliptic islands surrounded by a chaotic sea. The interest in periodic solutions to Hamiltonian systems has been unabated for three centuries now. In an oft-quoted sentence2 , Poincare considers them to be "the only opening through which we can force our way into an otherwise impenetrable citadel" .

!p2 -

1

2

"Mathematics and the unexpected", Chicago University Press, 1988. "Methodes nouvelles" , t. 1, p. 82.

viii

Introduction

In the case of completely integrable systems, periodic solutions are found by inspection. For nonintegrable systems, such as the three-body problem in celestial mechanics, they are found by perturbation theory: there is a small parameter € in the problem, the mass of the perturbing body for instance, and for € = 0 the system becomes completely integrable. One then tries to show that its periodic solutions will subsist for € -# 0 small enough. Poincare also introduced global methods, relying on the topological properties of the flow, and the fact that it preserves the 2-form L~=l dPi 1\ dqi' The most celebrated result he obtained in this direction is his last geometric theorem, which states that an area-preserving map of the annulus which rotates the inner circle and the outer circle in opposite directions must have two fixed points. And now another ancient theme appear: the least action principle. It states that the periodic solutions of a Hamiltonian system are extremals of a suitable integral over closed curves. In other words, the problem is variational. This fact was known to Fermat, and Maupertuis put it in the Hamiltonian formalism. In spite of its great aesthetic appeal, the least action principle has had little impact in Hamiltonian mechanics. There is, of course, one exception, Emmy Noether's theorem, which relates integrals ofthe motion to symmetries of the equations. But until recently, no periodic solution had ever been found by variational methods. This was because the least action principle is in fact a misnomer. Periodic solutions do not minimize the action; they are just extremals, i.e. curves for which the first-order variations vanish. So the classical methods of the calculus of variations, which are used for instance to find minimizing geodesics, do not apply to this case, and progress on this question had to wait till suitable mathematical tools had been developped. It is one of the achievements of modern nonlinear analysis that it is now possible to find periodic solutions of Hamiltonian systems by variational methods. Since the pioneering paper of Paul Rabinowitz in 1978, this problem has served as a trying ground for the various methods of critical point theory in function spaces, Morse and Liusternik-Schnirelman theory, linking and saddlepoint theorems. It is also the hope of all of us who have been working in this field that the variational techniques which have been honed on finitedimensional Hamiltonian systems will prove instrumental in solving even more difficult problelllS of mathematical physics, such as the study of nonlinear waves and fields. This book is decidedly mathematical in character. Its aim is to give the state of the art concerning nonlinear systems with convex Hamiltonian, that is, when H(t,p,q) is convex in the (p,q) variables. Physically, such systems behave like nonlinear springs, and one will therefore expect the existence of nonlinear modes of vibrations - that is, periodic solutions of the corresponding equations... Mathematically, a device originating with Frank Clarke dramatically improves the situation and makes the problem amenable to by now classical techniques of minimization, mountainpass theorem, and Liusternik-Schnirelman theory. Many, if not most, impor-

Introduction

ix

tant Hamiltonians are non-convex, for instance Newton's, H(p, q) = ~p2 - II!II' Such problems with singular problems can also be treated by variational methods, and are the subject of much active research nowadays. I feel, however, that the unity and simplicity the book gains by focusing on convex Hamiltonians more than offsets the loss in scope. In addition, some of the most interesting results in the convex theory, for instance the multiplicity results in Chapter V, have no counterpart in the nOn-COnvex case. This book is written with two purposes in mind: (a) to serve as an introduction to the subject of convex Hamiltonian system, and more generally to nonlinear analysis and critical point theory. (b) to serve as a reference for the properties of the index of periodic solutions to convex Hamiltonian systems. The first task has been greatly facilited by the recent appearance of the excellent book by Jean Mawhin and Michel Willem3 . By giving a concise and complete account of critical point theory, they have enabled me to concentrate on the various aspects of index theory. The index, as we define it in this book, is an integer associated with a linear system on a time interval, provided the Hamiltonian is positive definite. Other definitions exist, valid for any quadratic Hamiltonian, but here again the convexity assumption enables us to give a purely analytic description, without having to go into the geometry of the symplectic group. The advantage of convexity are not limited to ease of exposition. In the final analysis, it is because the index is a non-decreasing function of the interval in the positive definite case (in other words, all conjugate points contribute positively to the index), that we are able to prove the minimal period result of Chapter N and the multiplicity theorems of Chapter V. No such results are known in non-convex situations. I have intended Chapter I to be a complete exposition of index theory. In so doing, I have encountered the need for a reliable exposition of Krein theory, adapted to my particular needs. This beautiful theory stems from the study of stability of linear Hamiltonian systems. It is, of course, the creation of Mark G. Krein, but went unnoticed in the West, so that Jiirgen Moser independently rediscovered it a few years later. So the first three sections of Chapter I contain an account of Krein theory, which will be used in later sections to prove iteration formulas for the index. Chapter II is a self-contained account of convex analysis, and gives the abstract duality principle which is the setting for the remainder of the book. The last Chapters, III, IV and V, contain our results on periodic solutions of nonlinear Hamiltonian systems. They can be read separately, provided the reader is prepared to refer back to Chapters I and II occasionally. The need for the index theory of Chapter I does not occur before Section 3 of Chapter IV. 3

"Critical point theory and Hamiltonian systems", Springer-Verlag 1989.

x

Introduction

Difficulty increases from Chapter III (fairly elementary; requires no critical point theory, only a good grasp of the functional analysis in Chapter II) through Chapter IV (fairly technical; relies on the mountain-pass theorem of Ambrosetti and Rabinowitz; full proofs are given) to Chapter V (relies heavily on Liusternik-Schnirelman theory in the context of group actions; technicalities are refered to the original papers). Chapters III and IV are mostly concerned with nonautonomous problems: the fundamental period T is prescribed. We show the existence of solutions with the given period T (fundamental mode), and also of solutions with higher period kT (subharmonics). Chapter V deals with autonomous problems: the energy of the system is prescribed, and we seek periodic solutions (of any period) lying on that energy level. The culminating point is theorem V.3.15, which shows that there are at least two such solutions on each energy level, and infinitely many in general (except in the trivial case where there is just one degree of freedom, n = 1). In each section, propositions are numbered sequentially, together with definitions, lemmas, corollaries and theorems. Within the same section, they are refered to by this number. If they are in another section, or another chapter, the number ofthe section, or the chapter, is recalled also. For instance, Lemma 1 means Lemma 1 in the present section, while Lemma 1.1 means Lemma 1 in Section 1 of the present chapter, and Lemma I. 1.1 means Lemma 1 of Section 1 in Chapter I. Formulas are numbered according to the same principles. The results in Chapters III, IV and V have been obtained in the ten years of intense activity which followed Rabinowitz's landmark result and Clarke's duality principle. These years have been made extremely gratifying by the openness and good fellowship which prevail among those who have concerned themselves with this theory. I wish to thank particularly my friends and collaborators, Antonio Ambrosetti, Nassif Ghoussoub, Helmut Hofer, Jean-Michel Lasry and Pierre-Louis Lions, who have taught me so much. Paris, March 1989

I. Ekeland

Chapter 1. Linear Hamiltonian Systems

1. Floquet Theory and Stability Consider a system of m linear equations with continuous T-periodic coefficients:

x=

(1)

M(t)x

where M(t) is a real m x m matrix, depending continuously on t E IR such that: M (t + T) = M (t) .

(2)

The solutions to the initial value problem: (3)

x=

M(t)x,

x(O) = ~ E IRm

are given by:

(4)

x(t) =

R(t)~

,

where the matrix R(t) solves the initial-value problem:

(5)

~ R(t) =

M(t)R(t),

R(O) = I .

We shall refer to R(t) as the matrizantofsystem (1). By the general theory of linear systems, the matrizant is invertible for every t, with R(t)-l = R( -t). In the case of a system with periodic coefficients, such as this one, Floquet theory gives us some more information. Indeed, note that if R(t) solves problem (5), then RT(t) := R(t+T) solves problem:

(6)

d dt RT(t) = M(t)RT(t),

so that RT(t) = R(t + T) = R(t)R(T).

RT(O) = R(T) ,

2

1. Linear Hamiltonian Systems

Since R{T) is invertible, the set of eigenvalues Spec R{T) does not contain O. Choose a simply connected domain n, and a determination of the logarithm, log: n -+ .. splits (non-uniquely) into a direct sum of irreducible invariant subspaces. We shall say that an eigenvalue A is semi-simple if each irreducible invariant subspace of E>.. is one-dimensional, that is, the invariant subspace is just the eigenspace:

(12)

E>.. = Ker (M - AI) .

M is diagonalizable if and only if all its eigenvalues are semi-simple. We now extend Proposition 2:

Proposition 5. Let A and J.l be eigenvalues of M. If Xli =I- 1, the associated invariant subspaces E>.. and E", are G-orthogonal.

10

I. Linear Hamiltonian Systems

Proof. Pick two vectors x E E).. and y E ElL" We have (M - AI)P x = 0 and (M - J-Ll)q Y = 0 for some integers 1 S p S m).. and 1 S q S mIL" Set p+q = m and argue by induction on m. For m = 2, we have p = q = 1, so that x and yare eigenvectors, and the result follows from Proposition 4. Assume the result holds for m < m and set m = m + 1. Define two new vectors x' := (M - AI) x and y' := (M - ILl) y. We have (M - AI)P-l x' = 0 and (M - /lI)q-l y' = o. It follows from the assumption that:

(13)

(Gx',y) = (Gx,y') = (Gx',y') =

o.

Replacing x' and y' by their values, we get three equations: (14)

(GMx,y) = A (Gx,y)

(15)

(Gx, My) = 71 (Gx, y)

(16)

(GMx, My) - A (Gx, My) -71(GMx,y)

+ A71(Gx,y) =

0.

Writing the two first equations into the third, and using the fact that M is G-unitary, we get:

(17)

(1 - A71) (Gx, y) So A71 = 1 or (Gx, y) =

=

0.

o.

o

Corollary 6. If Ais an eigenvalue of M away from the unit circle, the invariant subspace E).. is G-isotropic:

(18)

(Gx,y) =0

IAI -# 1, then

for all xandyinE)...

o

Proof. Take A = /l in Proposition 5.

Note that, if A = ±1, there are real eigenvectors, and they must be Gisotropic. We also have the following: Proposition 7. If A is an eigenvalue of M on the unit circle, and A is not semi-simple, then there must be a G-isotropic eigenvector. Proof. By the theory of the Jordan form, we can find an eigenvector x and another vector y E E).. such that: My = Ay+X. Hence: (GMx, My) = (AGx, AY + x) =

IAI2 (Gx, y) + A(Gx, x)

But the left-hand side is just (Gx, y) since M is G-unitary. Since we are left with (Gx,x) = 0, as desired.

IAI2 = 1, 0

2. Krein Theory and Strong Stability

11

We are now in a position to state the central definitions of Krein theory. Recall that a nondegenerate Hermitian form G can always be written as a sum of squares by a linear change of variables; the number p of positive squares and the number q of negative squares are invariant, that is, they do not depend on the particular basis in which G has been diagonalized. The pair (p, q) is called the signature of G. Definition 8. Let A be an eigenvalue of M on the unit circle. The restriction of G to the invariant subspace E).. must be nondegenerate. The signature (p, q) of G on E).. is called the Krein type of A. If q = 0, that is, G is positive definite on E).., we say that A is Krein-positive. If q = 0 or p = 0, we say that A is Krein-definite; otherwise, we say that A is Krein-indefinite.

Denote by G).. the restriction of G to E)... The fact that G).. is nondegenerate follows readily from the orthogonal decomposition

(19)

with F).. =

EB ElL

w/-)..

and the two subspaces E).. and F).. are G-orthogonal by Proposition 5. Note a simple lemma: Lemma 9. If A has Krein type (p, q), then>: has Krein type (q,p). As a particular case, if A = ±l is an eigenvalue, we must have p = q.

Proof. Since the real part of (Jx, x) is zero, we have (Jx, x) = - (Jx, x). Hence: (20)

(Gx, x) = -i(Jx,x) = -(Gx,x) .

If [6, ... ~ml is a G-orthogonal basis for E).., then ~1' •.. ~m] is a Gorthogonal basis for B>:, with (G~k' ~k) = - (G~k' ~k). The result follows for the case when A is not real. If A = ±1, the invariant subspace E).. is even-dimensional and real. Starting from a G-orthogonal basis [6, ... 6m], which we normalize by (G~k' ~k) = ±l, we get a new G-orthogonal basis ~1' ... ~2m] where the signs of all the squares have been changed. Since the number of positive and negative squares is an invariant, there must be equally many of them. 0 If A is Krein-definite, there can be no G-isotropic vector in E)... It then follows from Proposition 7 and the preceding remark that if 1 or -1 are eigenvalues, they must be Krein-indefinite, and so must be all the eigenvalues which are not semi-simple. These simple remarks lead us to a characterization of strong stability: Theorem 10. M is strongly stable if and only if it is stable and all its eigenvalues are Krein-definite.

Proof. Assume M is stable and all its eigenvalues are Krein-definite. I claim M is strongly stable. Otherwise, there would exist a sequence Mn of unstable symplectic matrices converging to M. Either Mn has an eigenvalue outside the

12

1. Linear Hamiltonian Systems

unit circle, or Mn has an eigenvalue on the unit circle which is not semi-simple. In either case, there is a G-isotropic eigenvector Xn:

(21)

and

IIxnll =

1

(22) Since An is a root of Det (Mn - zI), we can extract from the sequence An a subsequence converging to a root of Det (M - zI), that is, an eigenvalue of M. By compactness of the unit ball, the Xn also have a convergent subsequence; let x be its limit. Taking limits in the preceding equalities, we see that x is an isotropic eigenvector of M. This is impossible since all eigenvalues of Mare Krein-definite. So M must be strongly stable. Conversely, assume M is strongly stable. Then M is stable, so that all its eigenvalues lie on the circle and are semi-simple. For every eigenvalue A with positive imaginary part, we choose in the eigenspace Ker (M - AI) a G-Qrthogonal basis, say [6, ... , ~m], which we normalize by the condition (G~k' ~k) = ± 1. This is possible since G is non-degenerate. We then take ~1' ... '~m] as a basis for the conjugate eigenspace Ker (M - XI). If ±1 is an eigenvalue, the corresponding eigenspace is real and even-dimensional; we can choose its G-orthogonal basis to be [6, ... ,~m'~1' ... '~m]' with (G~k'~k) = 1 and (G~k'~k) = -1. Putting everything together, we get a G-orthogonal basis of eigenvectors [6,· .. , ~n' ~1' ... '~n] for (C2n. We have (G~k'~k) = - (G~k'~k)' and by rearranging the basis, we can always assume that (G~k' ~k) = 1 for 1 ::::; k ::::; n. Assume there is an eigenvalue A which is not definite. It must have two eigenvectors with opposite G-norms, say ~1 and ~1 if A = ±1, and 6 and 6 if A =I- ±1. Define a linear transformation M, by setting:

(23)

M,6

(24)

M'~l = A (~1 sinhE + ~1 cosh E)

=

A(6 cosh E+

~1 sinh E)

if A = ±1, and

M,6 M,6

(25) (26) (27)

M'~l =

(28)

M'~2 =

A (6 coshE + 6 sinh E) = A(6sinhE+6coshE) =

X(~1 coshE + ~2 sinh E) X (~1 sinhE + ~2 cosh E)

if A =I- ±1, and M, = M on the invariant subspace generated by the other ~k. By construction, M, is real (that is, M,1R?n C JR2n) , and symplectic, that is, (G~, () = (G M,~, M,(), as we readily check. On the other hand, 6 (if A = ±1) or 6 + 6 (if A =I- ±1) is an eigenvector of M" the corresponding eigenvalue being Ae', which is outside the unit circle if E > O. So M, is not

2. Krein Theory and Strong Stability

stable, and ME stable.

---+

M when

E ---+

13

O. This contradicts the fact that M is strongly 0

We draw an interesting cosequence: a "normal form" for strongly stable linear Hamiltonian systems. Proposition 11. The linear Hamiltonian system (1) is strongly stable if and only if there is a real T-periodic symplectic change of coordinates x = P(t)z which puts in the form i = J H' (z), where H (z) is the quadratic form given by:

(29) and where Qi + Qj =1= 2k7r IT for 1 ::; i, j ::; n and all k E 7l.. Then ei01T , ... , eionT , are the Floquet multipliers of positive type, repeated according to multiplicity. Proof. Assume R(T) is strongly stable, and write its eigenvalues as ei01T, ... ,eionT,e-iolT, ... ,e-ionT, the ones of positive type first. Since all the eigenvalues must be Krein-definite, Qi + Qj =1= 2k7rT for all i, j and k; in particular, Qi =1= k7r, so ±1 cannot be an eigenvalue. Choose as before a Gorthogonal, basis [6, ... , ~n' ~ l' ... , ~n] of eigenvectors, the first ones being of positive type, (G~k' ~k) = 1. Define a new basis [(1, ... , (2n] of ([j2n, consisting of real vectors, by:

(30) (31) We check that (J(k,(j) = 0 unless Jk-jJ = n, and (J(k,(k+n) =-1. Denote by Z the transition matrix from the canonical basis of lR2n to the (-basis, that is, the real matrix whose columns are (b ... , (2n. We have Z* JZ = J, so that Z is symplectic. We obtain from R(T)~k = ei(h~k (setting (h := QkT)

(32) (33)

+ sin ()k(n+k sin ()k(k + cos ()k(n+k

R(T)(k = cos (h(k R(T)(n+k = -

.

This means that the restriction of R(T) to the invariant subspace generated by (k and (n+k is a rotation of angle ()k: it can be written as e JTOk . So R(T) can be written as e JCT , where C is a real diagonal matrix in the (-basis, with eigenvalues Qk, 1 ::; k ::; n, as it was defined in formula (29). The symplectic change of variables x = P(t)z, with P(t) = R(t)Ze- JCt then brings the original system into the form i = JCz. Conversely, assume that the system can be brought into the above form by a T-periodic symplectic change of variables. Then it is obviously stable,

14

I. Linear Hamiltonian Systems

and the Floquet multipliers are the eiOkT • All we have to show is that the eigenvectors are Krein-definite. Denote by ek, 1 ::; k ::; n, the standard basis of lR,2n, set (k := P(t)ek, so that R(T) is written as e JAT in the (-basis. Note that the vectors:

(34)

c _~+k+i~ "1 (tt} = "AI (it) = -1. The eigenvalues can cross each other and continue their motion on the unit circle. In most cases, however, the eigenvalues will leave the unit circle and start moving on the negative real half-line, where they are related by >"n+1(t) = Al(t)-l. Eventually, they will both come back to -1 at some later time t2, where they will resume their motion on the unit circle, with the Krein-negative one now moving clockwise on the upper half-circle. It will then interfere with the remaining Krein-positive eigenvalues, and perhaps leave the unit circle again. All this can happen for a stable T-periodic system: the eigenvalues must only be back on the unit circle at the times t = kT, k E 71... To have a more complete picture, we now turn to Krein-indefinite eigenvalues. As the following results show, they immediately split up into Krein-definite eigenvalues, and eigenvalues which leave the unit circle. Proposition 4. Consider again the system (1), (2). The set of points t such that R(t) has a Krein-indefinite eigenvalue on the unit circle is discrete; that is, there are only finitely many such points in any bounded interval.

Proof. Suppose R(t) has a Krein-indefinite eigenvalue A E U. If A is not semisimple, apply Proposition 2.7. If A is semi-simple, the eigenspace Ker (M -A1) coincides with the invariant subspace Ker (M - A1)m, on which the Hermitian form G is indefinite. In either case, there is a G-isotropic eigenvector. Now argue by contradiction. Say there are infinitely many such points tk in a certain interval. Then there are sequences tk --+ t, with tk =I=- t, and ek E (C2n, with lIek II = 1, such that:

(20) (21)

R (tk) ek = Akek (Gek, ek) = 0 . Arguing by compactness, we may assume that ek

(22) (23)

R(t)e =

lIell =

--+

eand Ak --+ A with:

Ae 1.

Start from the relation:

Take the inner product of both sides with Jek. The left-hand side vanishes:

1. Linear Hamiltonian Systems

20

(R(t) (~k -~), J~k)

(25)

=

(~k -~, R(t)* J~k)

= (~k -~, J>'~k)

=

(~k -~, JR(t)-l~k)

= A (~k -~, J~k)

and we are left with:

(26) The right-hand side vanishes by relation (21) since G = iJ. Divide by ---t 0, and take limits. We get:

(t - tk) (27)

which contradicts the fact that A (to) is positive definite.

0

Corollary 5. Let A E U be an eigenvalue of R (to) with Krein type (Po, qo). Then there is some open interval N around to and some neighbourhood V of A in (C2n such that, if to =1= t E .N', R( t) has only Krein-definite eigenvalues in V n U. If t < T, the Krein-positive eigenvalues are on the negative side of AO in Vnu, while the Krein-negative ones are on the positive side; their positions are interchanged when t > T. Denote by Pt (resp. qt) the sum of the multiplicities of the Krein-positive (resp. Krein-negative) eigenvalues contained in V n U. Then Pt and qt are constant on each side of to; we set

(28) (29)

Po := Pt , qa+..- qt ,

qo := qt Pa+ '-p .- t

for t < T for t > T .

We have: (30)

Po+ - qo+

= Po -

qo = Po-- qo .

Proof. By Proposition 4, there is some open interval N around to such that, for to =1= t E .N', R(t) has only Krein-definite eigenvalues on U. Choose a neighbourhood V of A in (C2n such that A is the only eigenvalue of R (to) in V. Remember now that the eigenvalues are the roots of the algebraic equation Det (R(t) - wI) = 0; taking a smaller N if need be, we may assume that the sum of the multiplicities of the eigenvalues of R(t) contained in V is Po + qo, and that they all converge to A when t ---t to. By Proposition 2, we know that Krein-positive eigenvalues move positively as t increases: since they converge to A, they must be on the negative side of A for t < T and on the positive side for t > T. The Krein-negative eigenvalues lie on the other side of A. So, for t =1= T, Krein-positive and Krein-negative eigenvalues cannot interfere. They have to stay on the unit circle, and their number stays constant as long as t =1= T. Everything is now proved except formula (30). Denote by E (to) the invariant subspace of R (to) associated with the eigenvalue A, and by E(t), for t E.N', the direct sum of the three following subspaces associated with R(t):

3. Time-Dependence of the Eigenvalues of R(t)

21

E+(t) is the direct sum of the invariant subspaces associated with the Krein-positive eigenvalues in V n u. E_(t) is the direct sum of the invariant subspaces associated with the Krein-negative eigenvalues in V n U. Eo(t) is the direct sum of the invariant subspaces associated with the eigenvalues in V which lie away from the unit circle.

When t ---7 to, the subspace E (t) converges to the subspace E(to). The Hermitian form G is nondegenerate on E (to), with Po positive squares and qo negative ones. If t is close enough to to, that is, if s has been chosen small enough, the restriction of G to E(t) will be nondegenerate with signature (Po, qo). The subspaces E+(t), E_(t), Eo(t) are G-orthogonal. We have Pt = dimE+(t) and qt = dimE_(t). I claim that the restriction of G to Eo(t) has signature (rt, rt) where 2rt = dim Eo(t). It then follows that: (31)

Po = Pt

+ rt

and qo = qt

+ rt

and we get formula (30) by substraction. Note first that, by Proposition 2.5, the restriction of G to Eo(t) is nondegenerate. Now approximate R(t) by a sequence Mk ---7 R(t) of diagonalizable symplectic matrices. Then Eo(t) is the limit of invariant subspaces Ek associated with Mk. Each Ek splits into one-dimensional eigenspaces corresponding to eigenvalues A of Mk away from the unit circle. Group these eigenspaces in pairs FA := EA EEl El/"Xi the FA are pairwise orthogonal by Proposition 2.5, and they contain isotropic vectors by Corollary 2.6. The restriction of G to FA must therefore have signature (1,1). So the restriction of G to Ek must have signature (rt, rd, where rt is the number of such pairs FA. Since Ek ---7 Eo(t), we have 2rt = dim Eo(t), and since the restriction of G to Eo(t) is nondegenerate, it must also have signature (rt, rt). D In other words, Krein-positive and Krein-negative eigenvalues leave the unit circle in pairs, while the remaining ones continue their motion on U in the direction prescribed by their Krein-sign. A Krein-indefinite eigenvalue is the place where such a collision occurs. We formalize this idea by a definition: Definition 6. The number rt is constant on each side of to:

(32) (33)

ro : = rt

for t < T

+·. -- r t ro

for t > T

We refer to 2ro as the number of eigenvalues which arrive on the unit circle at A, and to 2rt as the number of eigenvalues which leave. D Denote by m the multiplicity of the eigenvalue A. It follows from formulas (30) and (31) that:

(34)

2ro

+ Po + qo

= m

and 2rt

+ pt + qt =

m

22

I. Linear Hamiltonian Systems

•

t

T

Fig. 3. At time t = T, there is a collision between four simple eigenvalues of R(t) indicated by arrows resulting in an eigenvalue A on the unit circle with multiplicity 4 and Krein type (2,2). In the notations of Corollary 5, we have pt = qt = Po = Qo = 1, and Tt = Po - pt = 1 = Po - Po = TO· By Definition 6, this means that 2 eigenvalues arrive on the unit circle at A, and 2 eigenvalues leave.

(35)

Po

= Po +

ro

and qo

= qo + ro

(36) Later on (Proposition 5.13), we shall prove that

(37)

r;

+ ro

= m - dimKer

(R(T) -)..1)

We conclude this section by a general result, which ensures that eigenvalues of R(t) have to move with t. It is a consequence of Propositions 4 and 2, but we prefer to state it independently.

Proposition 7. Consider again system (1), (2). Let>. E U be an eigenvalue of R (to). There is some open intervalN around to such that, if to t=- t E.AI, then >. is not an eigenvalue of R( t).

Proof. Argue by contradiction. Say there are sequences hk and ~k E 1, then there is an co > 0 and a holomorphic map cp from a neighbourhood of 0 in CC to a neighbourhood of ei80 , with cp'(O) =1= 0, such that Det (R (11-, T) - cp(z)I) = 0 whenever 111- - 11-01 < co and zd = (11- - 11-0). It follows that the set of W E CC such that Det (R (11-, T) - wI) = 0 for some 11- E IR contains d real analytic curves WI(S), ... Wd(S), defined for lsi small, with Wj(O) = coi80 for 1 ~ j ~ d; one of them has to lie on the unit circle, say WI(S) E U for all s, and the angle between the tangents to Wj(s) and Wj+I(S) at 0 is 2n/d. Unfortunately, all this analysis has been carried out under the assumption that everything depends analytically on the parameter A, which rules out the possibility of studying time-dependence by this method (that is, taking T as parameter). This is why the results in this section are not covered directly by those we just quoted. However, the spirit is the same. For every d-dimensional irreducible invariant subspace of R(t), associated with the eigenvalue A E U, we will get (d - 1) curves bifurcating from the unit circle at A, each of which carries a pair of eigenvalues of R(t). The total number of such curves is the counts the "outgoing" curves (those which carry a pair of eigenvalues of R(t) for t 2: T; they start at A for t = T and separate for t > T, one inside the unit disk and one outside), and the "ingoing" ones.

rt +ro;

rt

ro

4. Index Theory for Positive Definite Systems We continue our investigation ofT-periodic linear Hamiltonian systems in IR2n : (1)

x=

JA(t)x ,

where A*(t) = A(t) = A(t + T), under the added assumption that the matrix A(t) is positive definite:

24

1. Linear Hamiltonian Systems

(A(t)x, x) > 0 Vx =J 0 .

(2)

Since A depends continuously on t, we can find positive constants a and b such that:

(3)

allxl12 2: (A(t)x,x) 2: b- I

llxl1 2

V(t,x)

and A(t) has an inverse B(t) := A(t)-I, which is also continuous, T-periodic and positive definite. By the preceding inequalities, we have:

(4)

b IIxl12 2: (B(t)x, x) 2: a-I

IIxl1 2

V(t, x) .

With every s > 0 we associate a real Hilbert space Es and a quadratic form Qs on Es defined by:

(5) (6)

Es:= {x E W I ,2 (0,s;lR2n ) Ix(O) = x(s)}

Qs(x, x)

:=

~

2

1 8

[(Jx, x)

+ (B(t)Jx, Jx)] dt

.

0

Note that, because of the periodicity condition on x:

(7) the true variable in formula (6) is x, and we can choose for x any primitive we like. The only prerequisite on x is that:

(8)

and

is x

dt

= 0 .

In other words, we have

(9) where qs is a quadratic form on the Hilbert space L~(O,s) defined by:

(10)

(11)

L~(O,s) = {u E L2 (0,s;lR2n ) qs(u, u)

liS

= -

2

[(Ju, lIsu)

11

8

udt =

o}

+ (B(t)Ju, Ju)] dt

.

0

Here lIs denotes the primitive of u with mean value zero:

(12)

4. Index Theory for Positive Definite Systems

25

Lemma 1. IIs is a compact operator from L~(O, s) into itself. We have II; = -lls and:

IIIIs I :S

(13)

s 27r .

Proof. IIs sends L~(O, s) into W 1 ,2(O, s) and the identity map from W 1 ,2(O, s) to L~(O, s) is compact by the Rellich-Kondrachov theorem. To check that it is antisymmetric, we just integrate by parts:

and the last term vanishes since:

(15) Estimate (13) is the so-called Wirtinger inequality, which we now prove. Take any u in L~(O, s). Expand u and IIsu in Fourier series, remembering that there is no constant term since their mean value in zero:

(16)

u(t) =

L une2i7rnt/s ,

n#O

(17)

II u(t) = ' " _8_ U e2i7rnt/s s L...t 2i7rn n n#O

with

,

U- n

= Un .

Using Parseval's equality twice:

IIIIsul1

2

=

L 4T2lunl n#O n 82

2 8

7r

(18)

:S

L 42lunl n#O 82

7r

2 8

=

82

42 Ilull

2

7r

D

So qs, as defined by formula (11), can be looked upon as a compact perturbation of a positive definite quadratic form. As such, it need no longer be positive definite or even nondegenerate, but the next result will show that it is positive definite on a subspace of finite codimension. Proposition 2. There is a splitting:

(19) such that: (a) E+(O,8), Eo(O,8) and E_(O, 8) are qs-orthogonal

(b) qs(u,u) > 0 Vu E E+(O,8) \ {O} (c) qs(u,u) = 0 Vu E Eo(O, 8)

26

I. Linear Hamiltonian Systems

°

(d) qs(u,u) < \fu E E_(O,s) \ {a} (e) Eo(O,s) and E_(O,s) are finite-dimensional.

Proof. Define a self-adjoint operator Bs from L~(O, s) into itself by:

(20)

(Bsu,u)

= is (B(t)Ju,Ju)

dt.

By formula (4), we have

(21) So Bs is an isomorphism, and (Bsu, u) defines a Hilbert space structure on L~(O, s) which is equivalent to the standard one. Endowing L~(O, s) with the interior product (Bsu,u), and applying to Jils the spectral theory of compact selfadjoint operators on Hilbert space, we see that there is a basis en, n E 1N, of L~(O, s), and a sequence An -+ in ill, such that:

°

(22)

(BSei,ej)

(23)

(Jilse n , u)

= 8ij = (BsAne n , u) Vu E L~(O, s)

.

Hence, expressing any vector u E L~(O, s) as u = E~nen' the expansion:

qs(u,u)

= 21 (-JIIsu,u) + 21 (Bsu,u ) 1

1

00

00

= -2 LAn~~ + 2 L~~

(24)

n=l

1

= 2 L (1 -

°

00

n=l

n=l

An) ~~ .

Since An -+ when n -+ 00, all the coefficients (1 - An) are positive except a finite number. Hence the result, with:

(25)

E+(O, s)

(26)

Eo(O,s)

(27)

E_(O,s)

° = {L~nenl~n = ° = {L~nenl~n = ° = {L~nenl~n =

if 1-An :S;0} if 1-An #0} if 1-An 2: 0 }

0

Definition 3. The index of system (1) on the interval (0, s), denoted by is, is the dimension of E_(O, s). The nullity, denoted by Vs, is the dimension of Eo(O, s). 0 The rest of this section will develop the meaning of these definitions. The nullity, to begin with, has a ready interpretation. The subspace Eo(O, s) is simply the kernel of qs; if it is non-trivial, that is Eo(O, s) # {O}, we say that qs is degenerate, and we have:

4. Index Theory for Positive Definite Systems

27

Theorem 4. The nullity Vs is the number of linearly independent solutions of the boundary-value problem

{± =

(Ps )

JA(t)x . x(o) = x(s)

o

Proof. Call1R2n the subspace of constant functions in L2 (0, Sj 1R2n): it is the orthogonal complement of L~(O, s). We have:

(28)

liS

qs(U'V)=2

0

(-JIIsu-JB(t)Ju,v)dt.

The kernel of qs consists of all u E L~(O, s) such that this interior product vanishes for all v E L~(O, s). This means that there is some constant ~ E 1R2n such that:

(29)

-JIIsu - JB(t)Ju =

~ .

This we rewrite as: (30)

IIsu -

J~

= -B(t)Ju

and since B(t) is the inverse of A(t):

(31)

-Ju = A(t) (Ilsu -

J~)

Now define x to be IIsu - J~. Then x(O) = x(s), and Eq. (30) reads as -J± = A(t)x, which is precisely system (1). 0 In other words, Vs is the dimension of Ker (R(s) - I), where R(t) is the matrizant of system (1). To say that qs is degenerate means that 1 is an eigenvalue of R(8). By Proposition 3.7, this can happen only at isolated points.

Definition 5. If Vs '" 0, we say that 8 is conjugate to 0 with multiplicity Vs. 0

We can also define conjugacy between any two points: 81 and 82 are conjugate ifthe equation ± = JA(t)x has a non-zero solution with x (81) = x (S2). It should be noted that this relation is not transitive: if (0,81) and (81, S2) are conjugate times, then 82 need not be conjugate to O. It should also be noted that the fact that 8 is conjugate to 0 does not imply that (8 + T) has the same property, even though A(t) is T-periodic. We are now ready to interpret the index in geometrical terms: Theorem 6. The index is of system (1) on the interval (0, S) is equal to the number of 8 E (0, S) which are conjugate to 0, each counted with multiplicity:

(32)

o The proof proceeds through a sequence of lemmas.

28

1. Linear Hamiltonian Systems

Lemma 7. When

°
(1, we have lis = 0, and hence by Lemma 8:

°

(40) Lemma 10. There are only finitely many points conjugate to interval (0,8).

°

o

in any bounded

4. Index Theory for Positive Definite Systems

29

Proof. It follows from Lemma 9 that each conjugate point 0' in (0, s) contributes at least Vcr to is. Since is < 00, there must be finitely many 0'. 0 The next - and final - lemma studies more closely the discontinuities of the function s -+ is. Lemma 11. The function s

-+

is is left continuous, and

(41)

Proof. We are dealing with a variable quadratic form qs on a variable space s). The first thing to do is to rescale everything to the fixed time interval

L~(O,

(0,1). Define a map p : L~(O, s) -+ L~(O, 1) by (PU)(t) = u(st). Then qs(u, u) = sq;(pu,pu), where q; is the quadratic form on L~(O, 1) defined by:

(42)

q;(v) =

111

2

0

[s (Jv, lllV)

+ (B(st)Jv, Jv)] dt

.

q;.

So is is also the index of Arguing as in Proposition 2, we can find a basis ei, i E lN, of L~(O, 1) and a sequence Ai in JR, with Ai -+ 0 when i -+ 00, such that:

(43) (44)

Here Bl(s) is the self-adjoint isomorphism from L~(O, 1) into itself defined by:

(45)

(Bl(S)U, v)

=

11

(B(st)Ju, Jv) dt .

Clearly B1 depends continuously on s in the operator norm. Writing u

=

E~iei, for any u E L~(O, 1), we get

(46)

q;(u,u)

=

~

f

(1- SAn~; .

i=l

Fix some point 0' > O. For the sake of convenience, we set ia = K. This means that there is some 0" > 0' such that is = K for all s in the interval I := (0',0"). By Lemma 10, we may assume that I contains no point conjugate to 0, so that for every s E I, we have Vs = 0, and the K first eigenvalues of qs are negative:

(47)

1- sX: < 0

for 1

~

i ~ K .

Fix i ~ K. By the preceding inequality, Ai ;:::: 1/ s > 1/0" so that the Ai are bounded away from zero for s E I. Because of the orthonormality relation

I. Linear Hamiltonian Systems

30

(43), the ei are bounded in L~(O, 1). So the set {sei/Ails E 1} is bounded, and its image by the compact map lIt (see Lemma 1) must be precompact. In other words, there is a sequence sen) ---+ 0' with sen) > 0', and a vector w E L~(O, 1) such that:

(48) Rewrite Eq. (44): ,s(n»)-lil1 esen) -_ s (n ) ( Ai i

(49)

Bl (s (n )) esen) i

.

Since the left-hand side converges to w, so does the right-hand side. But ---+ Bl(O') by continuity, and so:

Bl (s(n)) (50)

Since (Bl (s(n))e:(n),e:(n))

= 1, we get by continuity (Bl(O')ei,ei) = 1.

It follows that w = Bl(O')ei is non-zero, and hence, by formula (49), that A:(n) converges to some Ai. Taking limits in Eq. (49), we get: (51) with: (52) We proceed likewise for i = 1, ... , K. We get in this way K eigenvectors

ei, associated with K eigenvalues Ai, satisfying (51) and (52). Taking limits in formula (43), when s ---+ 0', we see that (B(O')ei' ej) = bij; as a consequence, the ei are distinct and non-zero. It follows from inequality (52) that q~ is negative semi-definite on the K -dimensional subspace generated by ei, ... , eK. Hence: (53) Lemma 9 provides us with the reverse inequality, so that (54) On the other hand, L~(O, 1) contains a subspace Eu of dimension iu on which qu is negative definite. Since qs depends continuously on s, the restriction of qs to Eu will still be negative definite when s is close to 0'. Hence iu :S i u - o. The converse inequality holds by Lemma 8, so

(55) and is is left continuous. Replacing iu by iu-o in formula (54), we get the result.

4. Index Theory for Positive Definite Systems

31

Proof of Theorem 6. The function s -+ is is integer-valued, left continuous and non-decreasing on (0, +00). Its value at any point S must therefore be equal to the sum of the jumps it incurred in (0, S). By Lemma 11, this is precisely the sum of the v s , for 0 < s < S. Theorem 4 and 6 provide us with a satisfactory interpretation of the nullity and the index. The next question is whether these numbers are actually nonzero, that is, whether points conjugate to zero exist in general. We begin by a remark about ordering. If Al and A2 are two symmetric matrices, Al A2 means that A2 - Al is positive semi-definite:

:s

(56)

for all x.

x = JAI(t)X and x = JA 2(t)x, with Al (t) and A2(t) continuous, symmetric and positive definite. Assume:

Proposition 12. Consider two linear systems

(57) Then their indices i! and i~ satisfy:

(58) Proof. Clearly BI(i) 2: B 2(t) for all t, so that the associated quadratic forms q; and q; satisfy q;(u,u) 2: q~(u,u) for all u in L~(O,s). So, q~ must be negative definite on any subspace where is negative definite, and formula (58) follows from the definition of the index. 0

q;

This will enable us to compare the system we are investigating, x = JA(t)x, with constant-coefficients systems, using estimate (3), which states in effect that: there are constants a 2: b- l > 0 with

aI2: A(t) 2: b- l I

(59)

for all t.

We denote by E[a] the integer part of the real number a, defined as follows:

(60)

E[a]

= k {:? k < a

:s k + 1 .

Note that this is not the standard definition: we get E[a] integers a. Lemma 13. Fix c' > coefficients:

(61)

o.

The index of the linear Hamiltonian system with constant

x =cJx

on the time interval (0, s) is given by:

(62)

= a-I for all

32

I. Linear Hamiltonian Systems

Proof. The solution of Eq. (61) is x = ecJtx(O). The only times conjugate to 0 are the Sk = 2k7r / C, k E 7L, each one with multiplicity 2n. The result follows from Theorem 6. D

We can now estimate the distance between two conjugate points, and the index over an interval. Proposition 14. Consider the Hamiltonian system

x=

(63)

JA(t)x

which satisfies condition (3). (a) Take any Sl, and let S2 > Sl be the first time which is conjugate to Sl. In other words, there is no s in the time interval (Sl, S2) which is conjugate to Sl. Then: (64)

(b) For any s > 0, we have the estimate: (65)

2nE [2:b]

~ is ~ 2nE [;;]

Proof. (a) There is no loss of generality in taking systems:

81

= O. Consider the three

x=aJx x = JA(t)x x = b- 1 Jx

(66)

(67) (68)

and denote by 1,s, is and is their indices on the interval (0,8). By Proposition 12, we have: (69) First take 13 gives:

8

= S2. Then is = 0 by Theorem 6, so that is = 0 and Lemma

(70) Then take s > Lemma 13 gives:

(71)

S2.

Then is ~ 1 by Theorem 6 again, so that 1,s ~ 1 and a 27r

-s>

1.

Letting s ---> S2, we get formula (64). (b) Follows from formula (69) and Lemma 13. We will show in the next section that From formula (65) it will then follow that:

8- 1 i s

D

has a limit when

8 ---> 00.

4. Index Theory for Positive Definite Systems

33

1.1m s- 1·~ < -an . -n < 8-+00 8 1f

(72)

1fb -

Notes and Comments. There are several ways to define the index of a linear Hamiltonian system. The index was first discovered and studied within the Russian school of ordinary differential equations, mostly for the needs of stability theory. After the pionneering work of Krein ([Krel) to [Kre4)), there was the fundamental paper by Gelfand and Lidsky [GelL). Again we refer to the treatise [YakS); a concise account appears in [Lev). In this approach, the index is understood as a winding number: the group of symplectic matrices M is shown to be homotopic to 8 1 , and the index of the system x = J A(t)x over the interval [0, s) just counts the number of times R(t) (or rather, its image in the homotopy) runs along the circle when moving continuously from R(O) = I to R(s). The direction ofrotation has to be taken into account, and therefore this index is a relative integer. Call it the standard index: istandard E 71.. It is the one that Amann and Zehnder [AmaZl]' [AmaZ2) and then Conley and Zehnder [ConZl], [ConZ2], [ConZ3) have been using in their celebrated work. It is of course related to the work of Maslov, which was going on at the same time [Mas], and in fact the standard index can be understood as a Maslov index. This aspect of things was discovered independently by Bott [Botl], and investigated by Duistermaat [Dui). The idea is to double the dimension. Consider the system: (Xl,X2) = (A(t)Xl'O) in lR2n

lR2n , endowed with the symplectic structures [~, 7]) = (J6, 7]1) (J6,7]2). The diagonal ~ = 7] then is a Lagrangian 2n-plane, Ilo say, and so is its image Ilt by the matrizant R(t) x I of this system. Now there is an intersection number for Lagrangian 2n-planes in symplectic 4n-space, which is zero if the two planes are in general position, and a relative integer otherwise. The index of the system x = J A (t ) x on the interval (0, s) is the (algebraic) number of times Ilt intersects Ilo with < t < s. It is the same as the standard index. The index of a positive definite Hamiltonian system, as defined in this section, was introduced by the author in the papers [Eke8) and [Eke9). In contrast with the standard index, it is a Morse index, and therefore a nonnegative integer. It was proved by Brousseau [Bro) that X

°

Morse index

= standard index - n .

Theorem 6 was proved in [Eke8) (see also [EkeH)). It is of course inspired by Morse's celebrated index theorem in Riemannian geometry (see [Mil)). The other results in this section come from [Eke9).

I. Linear Hamiltonian Systems

34

5. The Iteration Formula We are dealing with the linear Hamiltonian system

x=JA(t)x

(1)

in m2n

where A*(t) = A(t) is a continuous T-periodic positive definite matrix:

a IIxll2 2:: (A(t)x, x) 2:: b- 1 IIx211 A(t+T) = A(t) .

(2) (3)

with a 2:: b- 1

>0

It is natural to single out for consideration the indices iT, i2T, ... corresponding to intervals which are multiples of the basic period T. In this section, we shall compute ikT in terms of iT and the Floquet multipliers of system (1). We begin by a general inequality relating ikT to iT and I/T. Recall that the matrizant of system (1) is denoted by R(t), and that:

(4)

I/T

= dimKer (R(T) - I)

Theorem 1. For all k E 1N, we have:

(5) Proof Recall Proposition 4.2 and Definition 4.3 for the index and the nullity. Using the same notations, we see that there is a qT-orthogonal splitting:

L~(O, T) = E+ (0, T)

(6)

E9 Eo(O, T) E9 E_ (0, T)

with:

(7) (8)

dimE_(O,T) = i1 and q1(U) < 0 for 0 f:- U dim Eo(O, T) = 1/1 and Eo(O, T) = Ker q1 • Denote by p : L~(O, T)

---+ L~(O,

E

E_(O,T)

kT) the extension operator (see Lemma

4.8):

(9)

(pu)(t) =

and by r : L2(0, T)

(10)

---+

{~(t)

if O~t~T if T < t ~ kT

L2(0, kT) the translation by T:

(rw)(t)

:=

pw(t - T) .

Now define in L~(O, kT) subspaces Mi and N j by (note the different range of the index)

(11) (12)

N j = rjpEo(O,T) Mi = ripE_ (0, T)

for 0 ~ j ~ k - 2 for 0 ~ i ~ k - 1 .

5. The Iteration Formula

35

The subspaces are mutually orthogonal. Setting (13)

M = E9Mi

and

N = E9Nj j

we check easily that qkT is negative definite on M, and vanishes on N. In other words, N is a qkT-isotropic subspace, and any qkT-isotropic vector in M EB N must belong to N. Counting dimensions, we find:

(14)

dimME9N

= (k -1)vT + kiT.

We claim that N n KerqkT = {O}. Indeed, by Theorem 4.4, the kernel of qkT consists of all w = X, where x is a kT-periodic solution of x = JA(t)x. It follows from this equation that w is continuous and w(t) =I 0 for all t, whereas all functions in N must vanish on the interval ((k - I)T, kT) by formula (11). Set K = (k-l)vT+kiT. We have found a K-dimensional subspace MEBN, on which qT is negative semi-definite, and which contains no zero-eigenvector. The rest of the proof consists in finding a subspace with the same dimension, on which qkT is negative definite. We proceed as in Proposition 3.2, that is, we endow L~(O, kT) with the Hilbert structure defined by B kT . Define a linear selfadjoint operator C by:

(15) C is just the gradient of qkT for the inner product have:

(B kTU, v).

That is, we

(16) Now consider the map 1 + hC. For Ihl sufficiently small, it is an automorphism. Denote the linear subspace (1 + hC) (M EB N) by (M EB Nh, and the unit sphere of L~(O, kT) by S

(17)

S:= {w E L 2 (0,kt)i (BkTW,W) =

I} .

We have

(19)

'Ph(W) := (w

+ hCw) (BkT(W + hCw), w + hCw) -1/2

For every wE S n (M EB N), we have

(20)

d~ qkT ('Ph(W), 'Ph(W)) Ih=O = ( BkTCW, d~ 'Ph(W) Ih=J Differentiating formula (19) with (BkTW,W)

= 1, we get:

36

I. Linear Hamiltonian Systems

d~ 'Ph(W)lh=O = c,w -

(21)

qkT(W, w)w .

Writing this into formula (20) yields: (22)

ddhqkT('PdW),'Ph(W))lh=O = (BkTc'W,c'W) -qkT(W,w)2. If qkT(W, w)

= 0, that is, if wEN n S, we have c'w i= 0 and hence:

d~ qkT ('Ph(W), 'Ph(w))lh=O > 0 .

(23)

So qkT(W, w) ~ 0 on (M EEl N) n S, and if qkT(W, w) = 0 the above inequality holds. Since (M EEl N) n S is compact, it follows that there is an 10 > 0 such that, whenever -10 < h,O, we have:

(24)

Vw E (MEBN) nS,

qkT('Ph(W),'Ph(W)) < o.

By formula (18), this means that:

(25) So qkT is negative definite on (M EEl N) h. Its index must therefore be at least K = (k - l)vT + kiT. D Inequality (5) will be extremely useful in the sequel. For the time being, we seek something more precise, that is, we want a formula giving ikT in terms of iT. Recall that ikT is defined to be the index of the quadratic form

(26)

qkT(U,U)

="2l1kT [(Ju,lIu) + (B(t)Ju,Ju)]dt 0

on the Hilbert space:

Alternatively, ikT is also the index of the quadratic form:

(28)

l1kT QkT(X, x) = [(Jx, x) 2 0

+ (B(t)Jx, Jx)] dt

on the Hilbert space

(29)

EkT

= (x E W 1 ,2 (0, kT; IR2n) I x(O) = x(T)}

Surprisingly enough, to compute ikT in terms of iT, we have to complexify the situation. With every w E U (unit circle in 0 so small that

(81) (82) and there are no eigenvalues of R(T) of the form woe iO , with 0 < () :::; E. Now R(T) is the limit of a sequence Mn of symplectic matrices with simple eigenvalues. By Proposition 1.4, there is a sequence An(t) converging uniformly to A(t) such that Rn(T) = Mn ~ R(T), where Rn(t) is the matrizant of the system x = J An(t)x. Each of these systems defines a map iT : U ~ IN. Clearly n can be chosen so large that:

(83) (84)

iT (woe it ) = iT (woe it ) iT (woe-it) = iT (woe-it)

and that there are exactly Po eigenvalues of Rn(T) on the arc woe iO , 0 :::; () :::; E, each ofthem simple. Applying the preceding result to the system x = J An(t)x, we get:

5. The Iteration Formula

45

(85)

Using the Eqs. (81), (82) and (83), (84), this gives: (86)

which is precisely formula (77) for the Krein type (Po, 0). Lastly, we deal with the case of Krein-indefinite eigenvalues. Again, choose E > 0 so small that (81) and (82) hold, and there are no eigenvalues of R(T) of the form w o ei8 , with 0 < () :s; E. By Corollary 3.5, if t =1= T but IT - tl is small enough, R(t) will have only Krein-definite eigenvalues on the unit circle. Take t < T for instance. Then there will be exactly p;; Krein-positive and q;; Krein-negative eigenvalues on the arc eie , -E < () < E (counting multiplicities). We have the relation: (87) On the other hand, since VT (w o ei8 ) and always provided IT - tl is small enough:

VT

(w o e- i8 )

(88)

jt (woe if ) = jT (woe if )

(89)

jt (woe-if) = jT (woe-if)

are zero, we have,

The result obtained in the Krein-definite case applies to jt. We get: (90)

Comparing (81), (82) with (88), (89), this becomes: (91) which is the desired result, since q;; - p;;

= qo - Po.

o

It turns out that 1 is also a point of discontinuity of jT on U, even if it is not a Floquet multiplier. This is due to the fact that, in the splitting (35), the subspace QJ2n, which is contained in the kernel of QkT' lies in E}: none of the E'T, for w =1= 1, contains constant functions. Proposition 10. Assume 1 is not a Floquet multiplier of system (1). Then:

(92)

Proof. We separate in qh a fixed and a variable part: (93) where qo is obtained by setting h = 0 in formula (60).

46

I. Linear Hamiltonian Systems

It will be convenient to write:

(94) where L is a bounded self-adjoint operator in L2 (0, T; (C2n) defined by

(95)

(Lu)(t) = -J

it

uds +

~J

iT

udt - JB(t)Ju(t) .

Split L2 (0, T; (C2n) into L~ EB (C2n, where (C2n is the subspace of constants and L~ the subspace of functions with mean value zero. Write u = U o + ~ the corresponding decomposition of a vector u in L2 (0, T; (C2n). We shall try to write qh (u, u) as a sum of squares:

(97)

Mh :=

-iT

JB(t)J dt - hT 2iJ .

1

Fig. 4. The Bott map j when 1 is not a Floquet multiplier. Then j is constant on the arcs (1, >'I), (>'1, >'2), (>'2, X2), (X2' Xl) and (Xl, 1) j the Floquet multipliers which are off the unit circle play no role. Note the discontinuity which occurs at 1, even though it is not a Floquet multiplier: j (1) = iT but j (e iO ) = iT + n.

The matrix Mh defines a Hermitian operator in (C2n. It is dearly invertible for Ihllarge. Let us proceed with the calculations:

5. The Iteration Formula

47

(98)

(99)

( :=

~ + M;;l

iT

Lu o dt .

The map (uo,~) --+ (u o,() clearly is a linear isomorphism of L~ X (C2n into itself, and we may therefore adopt (u o , () as a new coordinate system in L2(0,T;(C2n). It follows from formula (98) that the subspaces U o = 0 and ( = 0 are qh-orthogonal, so the index of qh must be the sum of the indices of its restrictions to these subspaces. Setting U o = 0, we get the Hermitian form! (Mh(, () on (C2n. For Ihl large, it will behave like -hT2iJ, which is nondegenerate and has index n. Setting ( = 0, we get the Hermitian form

(100)

.. (u., u.)

= ~ (Lu o , uo ) - ~Re

(IT

£00 dt, M;;:'

lT Lu dt) o

.

When Ihl --+ 00, M;;l --+ O. In the limit, we get! (Lu o, u o), which is the restriction of the Hermitian form qo to L~. Writing JoT U o = 0 in formula (95), we get:

and we recognize on the right-hand side the Hermitian form qT of formula (4.11). Its index is iT = jT(l) by definition. Since 1 is not a Floquet mutliplier, qT is nondegenerate (Theorem 4.4), and so the restriction of qh to L~ will also have index iT for large Ihl. D All that remains to do is to compute j (w o) when wo is a Floquet multiplier. Once this is done, the function ir : U --+ 1N is completely determined. Proposition 11. Let Wo E U be a Floquet multiplier of system (1), with Krein type (Po, qo). Let 2r;;- be the number of Floquet multipliers which arrive on the unit circle at Wo

(102)

jT (woe iO )

(103)

jT (woe- iO )

If Wo

(104)

= 1,

= jT (wo) + qo = ir (wo) + Po -

with multiplicity m o, we have Po

. (iO) JT e

. (1) + n + 2mo 1 = JT -

r;; r;; .

= qo =

!mo, and

r 0_ =.JT (-iO) e .

48

I. Linear Hamiltonian Systems

Proof. Choose 10 > 0 so small that there is no Floquet multiplier of the form woe i () with 0 < () :S Eo Then

(105) (106) Now apply Corollary 3.5 with A = woo For t E Sand t < T, there are p;; Krein-positive eigenvalues on the arc ei()w o, -10 < () < 0, and q;; Krein-negative eigenvalues on the arcei()w o, 0 < () < Eo We have by Definition 3.6:

(107)

r;;

= Po -

p;; =

qo - q;; .

We now pause to note that the theory developped in Section 4, can easily be extended to other eigenvalues than 1. We define an instant s to be wo-conjugate to 0 if the boundary-value problem in T so small that (a) we are in the situation of Corollary 3.5, that is the arceiOwo, with o < () < E, carries pt eigenvalues of R(t), (b) the interval (T, t) contains no point which is wo-conjugate to 0, (c) it (woe iE ) = iT (woe iE ). Since T is wo-conjugate to 0 with multiplicity do, we have, by a suitable variant of Theorem 4.6: (120)

On the other hand, since R( t) has Krein-definite eigenvalues only, we may apply Proposition 9, and we get: Jt. (woe iE) - Jt. (Wo ) -_ -Po+ .

(121)

Replacing the two terms on the left by their values, given by (c) and (120), we get: (122)

. JT

(woein)

-

. JT

(WO ) --

d0

-

Po+ .

Comparing with the value given for the left-hand side by Proposition 11, we get (123)

o

Now pt = Po - rt by definition. The result follows. Corollary 14. Assume Wo is semi-simple: mo = do. Then, if Wo (124) (125)

iT (woe iO ) = iT (wo) + qo iT (woe- iO ) = iT (wo) + Po

1- 1:

50

I. Linear Hamiltonian Systems

and ifwo = 1:

+ ~o . (-iO) . (1) + 2" mo . JT e =JT

(126)

iT (e iO ) =iT(l)

(127)

o

Proof. If follows from (119) that r;; = r;; = O. Write this into Proposition 11.

o

Corollary 15. For Wo =11, we have: (128)

Max: { -Po

+ do, qo - ~o}

:::; iT (woe iO )

(129)

Max: {-qo

+ do, Po - ~o}

:::; iT (woe- iO )

iT (wo) :::; qo

-

iT (wo) :::; Po .

For Wo = 1, we have (130)

o

Proof. By Proposition 13, we have r;; :::; mo - do. By formula (3.34) we have 2r;; :::; mo. We have r;; 2: O. Write all this information into Proposition 11. 0

We have now completely determined iT: we know its value at w = 1, namely iT, and the rules we have given determine its values for all wE U. Let us do the calculations in two important cases. Proposition 16 (Hyperbolic case). Assume there is no Floquet multiplier on the unit circle. Then: (132)

ikT = k (iT

+ n) -

o

n .

Proof. The function iT : U -+ JN has no point of discontinuity except w = 1. We have iT(l) = iT, and iT (e iO ) = iT + n by Proposition 10. Hence: (133)

iT(W) = iT + n

\::Iw =11

o

and Eq. (132) follows from Bott's formula (42).

Proposition 17 (Simple eigenvalues on the unit circle). Assume that there are 2N Floquet multipliers on U, all of which are simple, and none of which is a root of unity:

0< a v

(135)

Set aD = 0 and Wv-l and wv: (136)

1

< "2 1:::; 1/:::; N .

(134)

aN+! = ~.

J.v . = J·T

Define i V E JN to be the value of iT between

(e2i7rQ ) ,

'" L.(x) such that:

H'(x) = >.(x)G'(x) \/x E

(40)

2: .

Since neither H' nor G' vanish on E, and both are C 1 functions of x, the number >.(x) must be positive, and a C 1 function of x. Define the reparametriyat ion a of z by: d

(41)

dtO' = >.(z(t)) We get:

(42)

i.,.(s) = i(t) (dt/ds) = i(t)(dO'/dt)-1 = i(t)>. (Z(t))-1 = JH' (z(t)) >.(t)-1 = JG' (z.,.(s)) .

o

We have associated with z various numbers: its Floquet multipliers, its index and its mean index per period. Because of Proposition 12, we have to ask which of these numbers depend on the particular representation H we have chosen for E, and which ones do not. In other words, if we replace H by G and z by z.,., are of these numbers unchanged? It turns out that the Floquet multipliers and the mean index per period are unaffected, while the index changes; that is, iT(z; H) and is (z.,.; G) are different in general. Proposition 13. Let G : 1R2n

-+ 1R be a C 2 function satisfying conditions (30) to (33), an let a be defined as above. Then (T, z) and (S, z.,.) have the same Floquet multipliers with the same multiplicity and the same Krein sign.

Proof. Denote by 2

~

(3 .

o

In this statement, Ta is taken to be the minimal period of Za, but the same result holds of course for any period. Denote by ja(w) the Bott function of (Ta, za). We know from Proposition 6.13 that the Za, for 0: ~ 1, have the same Floquet multipliers with the same multiplicity and Krein sign. So the Bott functions coincide on the intervals of continuity, up to an additive constant which is determined by ja (e±O). We are now able to determine this value, at least in the generic case:

Proposition 6. Assume that 1 is a double Floquet multiplier. Then: (64)

(65)

(e±iO) = ja (e±iO) = jf3

+n+1 ja(1) + n if

jf3(l)

Proof. As we just mentioned, for all Floquet multiplier, we have:

0:

if 1 < (3 S; 2 0:

>2.

and (3, and for all wE U which is not a

(66) where c(o:, (3) is a constant depending on 0: and (3. On the other hand, we know that the mean index per period Ta does not depend on 0: (see Theorem 6.14). But Theorem 5.18 tells us that Ta is the mean value of the function ja on the unit circle; it follows that c( 0:, (3) == O. It then follows from formula (66) that, for all 0: and (3:

7. Examples

73

(67) On the other hand, if we take into account the identity j",(1) and the similar one for /3, Proposition 5 gives:

= iT (z",; H",),

(68) and formula (5.130), which is applicable since 1 is a double Floquet multiplier, yields:

(69)

n ::; j", (e±iO) - j",(1) ::; n + 1

(70)

n ::; j{3 (e±iO) - j{3(1) ::; n + 1 .

Putting the inequalities (67) to (70) together, we get (64) and (65) as the only possibility. Note that, for a = 2, this agrees with Corollary 5.14. 0

A General Result. We go back to the assumptions of Section 6. Start from a C 2 compact hypersurface E C IR2n; assume that E bounds a convex set with non-empty interior. Consider a C 2 function H : IR2n ---* IR, such that:

(71)

Vx,

(72)

H(x)

~

0

E = {xIH(x) = 1}

vx E E,

(73)

(74)

Vx E E,

H" (x) is positive definite.

Consider the Hamiltonian flow periodic solution z(t) on E:

(75) (76) (77)

H' (x) =I- 0

x

= J H' (x) associated with

H and a

i = JH'(Z) H(z) = 1 z(O) = z(T) .

Denote by iT(Z; H) the index of this periodic solution on the interval (0, TJ, and by i(T, z; H) := lim ikT(Z; H)jk its mean index per period.

Theorem 7. If n (78)

~

2, we have:

i(T,z;H) > 2.

Proof. By Theorem 6.14, we may assume that H is positively homogeneous of degree 2. By Lemma 3, Ker (R2 (T2) - I) contains at least two vectors, x(O) and X(O). So T2 is conjugate to 0 with multiplicity 2, and Theorem 5.1 implies that i(T, z; H) ~ 2. We want a strict inequality.

74

I. Linear Hamiltonian Systems

If 1 is a double Floquet multiplier, Proposition 6 gives:

(79) There are at most 2(n -1) more Floquet multipliers on the unit circle, to be placed symmetrically with respect to the real axis, and each of them can decrease h by its multiplicity at most (see Proposition 5.9). It follows that, for any w E U, we have:

h(w) 2 h(l)

(SO)

+n +1-

(n - 1) = j2(1)

+2 .

So j2 is a step function, which is bounded below by 2, and which takes a value strictly greater than 2 on a neighbourhood of 1 (except at 1 itself). Its mean value must therefore be strictly greater than 2, and formula (7S) follows from Theorem 5.1S:

(Sl) If 1 has higher multiplicity, some even number m 2 4 say, Proposition 5.11 gives

h (e±iO) = h(l) + n + m/2 - TO

(S2)

with 2To denoting the number of Floquet multipliers which arrive on the unit circle at 1, so that 2To :S m. There are at most (2n - m) more Floquet multipliers on the unit circle. As before, it follows that: (S3)

j2(W) 2 h(l)

+ n + m/2 -

TO - (2n - m)/2

=

h(l) + m - TO .

Hence:

(S4) So h 2 2 everywhere, and formula (S2) yields j2 2 n near 1. We are in the same situation as when m = 2. 0 Notes and Comments. The strict inequality i > 2 will be crucial far the multiplicity result of the last chapter (Corollary V.3.16). It was first proved in [EkeH3].

8. Non-periodic Solutions: The Mean Index Let H : rn?n ----t JR be a convex Hamiltonian of class C 2 . We will be working on a fixed energy level, H = 1 say. That is, we consider the set:

and we assume that it is compact, non-empty, with H'(x) non-vanishing and H"(x) positive definite for every x E E. It follows that E is a C 2 hypersurface in JR2n .

8. Non-periodic Solutions: The Mean Index

75

Denote by cpt the flow associated with the Hamiltonian system

± = JH'(x)'

(1)

and choose some

E

> 0 such that the set U:= {xll- E < H(x) < 1 +E}

(2)

is bounded, with H' (x) =I- 0 for all x E U. Then cpt : U all t. We have

-+

U is well-defined for

(3) by Liouville's theorem and

(4) since H 0 cpt = H. There is a unique (2n - I)-form

(5)

fJ,

Qn lR2n such that

dXl /\ ... /\ dX2n

= dH /\ fJ,

•

Applying (cpt)* to both sides, we get dH /\ fJ, = dH /\ (cpt)* fJ,. By the uniqueness of the decomposition (5), we conclude that fJ, itself is preserved by the flow:

(6) Denote by i E

:

E -+ lR2n the standard injection, and set

fJ, E :=

(i E) * fJ,.

Lemma 1. fJ,E is a nondegenerate (2n-l)-form on E which is preserved by the Hamiltonian flow cpt.

Proof. All we have to do is to prove that fJ,E is nondegenerate. This follows immediately from Eq. (5) and the fact that dH vanishes on TE. Indeed, if we could find independent vectors 6, ... , 6n-l in TxE such that fJ,E (6,··· j 6n-l) = 0, we would have

(7)

(dXl/\ ... /\ dX2n) (H'(x),6, ... ,6n) = 0

which is clearly impossible.

D

From now on, we will consider fJ,E as a finite positive measure on E, invariant by cpt. This is the setting for ergodic theory. From now on, a.e. will mean "almost everywhere with respect to fJ,E". Take some ~ E E, and consider the (not necessarily periodic) solution x(t) = cpt(~) of the Cauchy problem. For each T, the index iT(~) of the linearized system iJ = JH" (x(t)) y

76

I. Linear Hamiltonian Systems

on the interval (0, T) is well-defined, but there is no reason why iT(~)/T should converge to some limit - unless x(t) happens to be periodic. What we will show, however, is that this limit exists for a.e. ~ E E. Theorem 2. There is a borelian function I : E

10 cpt

(8) 1. T

(9)

-ZT --+

I

=I

--+

[0, (0) such that

Vt

when T

--+ 00

the convergence being L1 and a.e.

D

The proof consists in checking the assumption of Kingman's subadditive ergodic theorem. For each fixed T, consider the function iT : E --+ [0, 00 ). By definition, iT(~) is the index of the quadratic form

(10)

ql(u,u) = loT [(Ju,lIu)

+ (HII

(cpt(~))-1 Ju,Ju)] dt

on the space L~ (0,T;JR 2n ). It depends continuously on ~ E E. Arguing by compactness, as in Sect. 4, we find that iT is a lower semi-continuous function of ~, and hence that it is borelian. By Lemma 4.8, we have:

(11) To show that T-li T converges when T to show that N- 1iN converges when N --+ attention to the sequence iN, N E IN.

--+ 00,

00,

it is therefore sufficient

N E IN. So we restrict our

Lemma 3. The sequence iN is cp-subadditive, that is:

(12) Proof. Fix some ~ in E. By definition, L~ (0, N; JR2n) and L~ (0, K; JR2n) contain subspaces EN and E K , with dimension iN(~) and iK (cpN(~)), such that the restrictions of qt and q'kN (f,) to EN and EK are negative definite. Now imbed everything into L~ (O,N +K;JR2n ). More precisely, consider the subspace EN and E K defined as follows:

(13) (14)

EK -

EK

=

=

{u E L~(O,N + K)I UI(O,N) E EN and UI(N,N+K) = O} 2

{u E Lo(O, N

+ K)I

UI(O,N)

=

°

and UI(N,N+K) E EK} .

The quadratic form qt+K restricts to EN as qt, and to EK as q'kN (/;) They are orthogonal subspaces, both for the standard Hilbert structure on

8. Non-periodic Solutions: The Mean Index

77

L~ (0, N + K; IR2n) and for qh+K' If follows that the restriction of qh+K on EN EB E~+K is negative definite. Hence formula (12). 0 To apply Kingman's theorem, we have to show that the iN are integrable. In fact, they are uniformly bounded. Introduce the following definition: Definition 4. Let 8 and .1 be two be two positive numbers, with 0 < 8 :S .1. We say that H" is (8, .1 )-pinched on E if:

(15)

o

If H" is (8, .1 )-pinched on E, it will be (8', .1')-pinched for every 8' and .1' < .1. The best values for 8 and .1 are given by:

(16)

>8

28 = Min Spec H"(x) xEE

(17)

2.1 = Max SpecH"(x) xEE

both numbers being well-defined and positive, since H is C 2 , E is compact and H"(x) is positive definite for all x E E. Lemma 5. Let H" be (8, .1)-pinched on E, and take anye E E. We have:

(18) Proof. We simply apply Proposition 4.14 with a = .1 and b = 8- 1 . Recall that we have a non-standard definition of the integer part:

(19)

E[a] = k {::} k < a :S k + 1 .

o

We have thus found uniform bounds for iT. Theorem 2 now follows from: Kingman's Subadditive Ergodic Theorem. Assume cp : E -+ E is measurepreserving and iN E L1 is a cp-subadditive sequence such that:

(20)

I~f ~

J

iN

> -00

.

Then N- l iN converges a.e. and L1 to some cp-invariant function I such that (21)

o We get from Lemma 5 an additional estimate:

(22)

78

I. Linear Hamiltonian Systems

Notes and Comments. The mean index of a non-periodic trajectory is introduced here for the first time. It shall not be used in the rest of the book, and the only purpose of this very short section is to show that it exists, and to ask what are its properties. All questions are open: we would particularly like to know (a) what are the relations, if any, between the mean index I(~) and the Liapounov exponents, as defined in the general theory of dynamical systems via Oseledec's noncommutative ergodic theorem (b) how representative the periodic points are: for instance, is I continuous at periodic points? In the case of an elliptic periodic point, with suitable generic properties, KAM analysis might throw some light on the matter. I learned Kingman's subadditive ergodic theorem from N. Ghoussoub.

Chapter II. Convex Hamiltonian Systems

1. Fundamentals of Convex Analysis We start from a duality pairing (X, X*, (.,.)), that is, two real vector spaces X and X*, and a bilinear map (x, x*) ---+ (x, x*) into lR which separates points: \Ix E X,:Jx* E X*:

(x,x*)

-=f=.

0

\lx* E X*,:Jx EX:

(x,x*)

-=f=.

0 .

We endow X and X* with the corresponding weak topologies, a (X, X*) and a(X*,X). A standard example of such a situation, and the only one which is of any use, is the case when X is a Banach space and X* its topological dual. If we do not take this as the starting situation, it is just to stress the fact that everything in this section depends on the weak topologies only. We will consider functions with values in lR U { +00 }. The introduction of +00 as an admissible value does not stem from a abstract desire for generality, but from very precise needs which will appear presently. Let F : X ---+ lR U { +oo} be such a function. We define its domain dom F to be the set of points x E X where F(x) is finite: domF:= {x E X I F(x) < +oo} and its epigraph epi F

c

X x lR as follows:

epiF = ((x, a) E X x lR I F(x) 2 a} Note that, if x rt domF, there is no point in epiF which lies above x. We shall say that F is proper if it is not identically +00, that is, if epi F is non-empty.

Definition 1. F : X convex set.

---+

lR U { +oo} is a convex function if epi F

c

X x lR is a 0

Definition 2. F : X ---+ lR U (+oo} is lower semicontinuous (l.s.c.) if epi F C X x lR is a closed set. 0

80

II. Convex Hamiltonian Systems

These definitions can immediately be translated into equivalent properties:

Proposition 3. F : X inequality holds:

----+

IR U {+oo} is a convex function iff the following

F (ax

+ (3y)

:::; aF(x)

+ (3F(y)

for all x and y in X, and all positive real numbers a and (3 such that a + (3 = 1.

o

Proposition 4. F : X ----+ IR U {+oo} is a l.s.c. function iff one of the following holds: (i) for every a E IR, the set {xIF(x) :::; a} is closed in X. (ii) lim infy-+x F(y) 2:: F(x). 0 As an example, consider the indicator function OK of a subset K eX:

OK(X)

=

{O+00

if x E K otherwise.

The function OK is convex iff K is convex, and it is l.s.c. iff K is closed. If F is convex (l.s.c.) so is >"F for all >.. > O. If F and G are convex (l.s.c.) so is F + G. The set of all convex (l.s.c.) functions on X is a convex cone (not a linear space: substracting is forbidden). There is one more operation which can be defined on convex functions: inf-convolution.

Definition 5. Let Fl and F2 be convex functions on X. Consider in X x IR the subset:

and define F : X

(1)

----+

IR U { -oo} U { +oo} by:

F(x) := inf {al (x, a) E epi Fl + epi F 2 }

If F(x) > -00 for every x E X, we say that F is the inf-convolute of Fl and F 2 , and we write:

o

We can write formula (1) in another way: (2)

So the inf-convolute of Fl and F2 is well-defined if the right-hand side of this formula is never -00. It is sufficient, for instance, that Fl and F2 both a bounded from below. Then F 1 0F2 is convex, and epiFl + epiF2 C epi (F1 0F2) C epiFl + epiF2 . Note that, in general, epi (F1 0F2) will not be equal to epi Fl +epi F 2 . This is because the infimum need not be attained on the right-hand side of formula

1. Fundamentals of Convex Analysis

81

(2): we construct epi (F1DF2) from (epiFl + epiF2) by "vertical closure", that is, taking the intersection with every vertical line {x} x JR in X x JR, which is either empty or an unbounded interval, and closing it if it is open. Note also that epi Fl + epi F2 need not be closed even if epi Fl and epi F2 both are. It follows that F1DF2 need not be l.s.c. even if Fl and F2 both are convex l.s.c. The main theoretical tool in the study of duality pairings (X,X*, (- , .)) is the Hahn-Banach theorem, and the separation theorems which follow from it. For instance, we know that every closed convex subset of X x JR is the intersection of all closed half-spaces which contain it. But epigraphs of convex l.s.c. functions are particular cases of closed convex sets, and every non-vertical closed half-space can be written as the set of (x, a) such that (x, x*) -a ::; m, for some x* E X* and mE JR. Note that the separating hyperplane a = (x, x*)-m then is the graph of the continuous affine functions x ~ (x, x*) - m. It turns out that vertical half-spaces can be neglected, and we get the fundamental result. Theorem 6. Let F : X ~ JR U {+oo} be a convex l.s. c. function. Denote by M(F) the set of its continuous affine minorants:

(3)

(x*,m) E M(F) {:} (x,x*) -

m::;

F(x)

"Ix EX.

Then F is the pointwise supremum of all functions in M (F):

(4)

F(x) = sup {(x, x*) - ml (x*, m) E M(F)} .

o

The next thing to do is clearly to gain some understanding of the set M(F). It is defined by formula (3) as a subset of X* x JR. Let us fix x* in X, and look at all the m E JR such that (x*, m) E M (F). Formula (3) gives:

(5)

(x*,m) E M(F)

{::=}

m 2: (x,x*) - F(x)

{::=}

m 2: sup {(x,x*) - F(x)}

"Ix E X

xEX.

If we now define

(6)

F* (x*) := sup {(x, x*) - F(x)} xEX

then formula (5) becomes m 2: F*(x*), that is, M(F) is simply the epigraph of F*. Note that, if F is proper, then the right-hand side of formula (6) is not -00, so that F* is well-defined as a function from X* into JR U {+oo}. Formula (4) now becomes: F(x) = sup {(x,x*) - mlx* E X*,m 2: F*(x*)} =

sup {(x, x*) - F*(x*)} x*EX*

We now summarize this discussion:

82

II. Convex Hamiltonian Systems

Definition 7. Let F : X -+ lR U {+oo} be a proper function. The function F* : X* -+ lR U { +oo} defined by

(7)

F*(x*) = sup {(x,x*) - F(x)} xEX

is called the Fenchel conjugate or Legendre transform of F.

o

Proposition 8. Let F be a proper convex function and F* its Legendre transform. We have:

(8)

L

F(x)=

{(x,x*)-F*(x*)}.

x'EX'

o

As an easy consequence offormulas (7) or (8), we have Fenchel's inequality (9)

F(x)

+ F*(x*)

~

(x, x*)

Vx E X Vx* E X* .

Proposition 9. If F is proper, F* is convex l.s.c. If in addition F is convex l.s.c., we have:

(F*r

(10)

= F .

o

Proof. Formula (7) defines F* as the pointwise supremum of a family of affine continuous functions on X*, namely x* -+ (x, x*) - F(x), indexed by x E X. It follows that F* must be convex l.s.c. Replacing F by F* in Definition 7, we get: F**(x):=

L

{(x, x*). - F*(x*)}

x*EX*

If F is convex l.s.c., the right-hand side coincides with F(x) by Proposition 8.

The Fenchel conjugation F -+ F* sends the set of proper convex l.s.c. functions on X onto the set of proper convex l.s.c. functions on X*; its inverse is itself. Therefore, any property of F must translate into some property of F* , even though the transformation may not be obvious. Note the following easy relations:

(11) (12)

F ::; G

{::==}

F*

(AFr (x*) = AF* (x;)

~

G*

for A> 0 .

On the other hand, the direct computation of (F+G)* is surprisingly difficult. We must use a detour.

Proposition 10. Let Fl and F2 be convex l.s.c. functions on X, such that FIDF2 is well-defined. Then:

1. Fundamentals of Convex Analysis

83

o

(13) Proof. By Definition 7:

(FI OF2 )* (x*)

= sup {(x, x*) x

(FI OF2 ) (x)}

Plug in formula (2):

(FI OF2 )* (x*) =

s~p {(X,X*) - Xl~~~=X {Fl(Xl) + F2(X2)}}

= sup {(Xl

+ X2, X*)

- Fl(Xl) - F 2(X2)}

= F{(x*) + F;(x*) . Let Fl and F2 be convex t. s. c.

o

Corollary 11. functions on X, such that Fl + F2 is proper. Then Fi OF2' is well-defined and:

o

(14)

Proof. Replace Fl by Fi and F2 by F2' in Proposition 10. We get:

(F{OF;)* = Fl + F2 .

o

Taking conjugates on both sides, we get formula (14).

We already know that FiOF2' is convex. If it also happened to be l.s.c., we could apply Proposition 9, and formula (14) would reduce to a much more pleasant identity, (Fl + F2)* = FiOF2'. More about this in the next section. Similarly, there are difficulties in finding (G 0 A)*, when A is a linear map. We shall allow A to be an unbounded operator. To be precise, A will map a linear subspace dom A c X into Y; we define

(15)

graphA: {(x,Ax)lx E domA} .

It is a linear subspace of X x Y. We say that A is closed if its graph is closed. The orthogonal of graphA in X* x y* is the graph of another closed operator -A* from domA* c y* into X*: graph - A* := (graphA)~ .

(16) In other words:

(17)

x*

+ A*y* = 0 ~

(x, x*)

+ (Ax, y*) = 0

'ix E domA .

If A is a closed operator from X to Y and G : Z ~ IRU { +oo} is a function on Y, we define GoA: X ~ IR U {+oo} as follows:

(18)

(GoA)(x)

G(Ax) if x E domA ={ +00 otherwise .

84

II. Convex Hamiltonian Systems

Proposition 12. Let A : dom A

---t Y be a closed operator, with dom A eX, and G : Y ---t 1R U {+oo} a convex l.s.c. function. Define a function P : X* ---t 1R U { +oo} by:

(19)

p(x*) = inf {G*(y*)IA*y* = x*}

the right-hand side being +00 if x* 0 such that, if Ilxll :S a, we can write x = Xl - X2 with Ft(XI) < 00 and F2(X2) < 00, so that the right-hand side of inequality (14) is finite. So the function G is finite on the ball Ilxll :S a; since it is convex and l.s.c., it must be continuous on the interior Ilxll :S a. It follows that it is uniformly bounded on some ball around 0:

(15)

3'1] > 0, 3c:

Ilxll:S 'I]:::} G(x) :S c .

Writing this into Fenchel's inequality (9), and remembering formula (10) as well, we get: (16)

IIxll:S 'I] :::} (x, y*) :S c + f + 1 .

(x*, y*) E.4 and

Replacing x by -x, we get the reverse inequality:

(17)

(x*,y*) E.4 and

IIxll:S 'I]:::} l(x,y*)I:S c+f+ 1.

We now use the ultrafilter:

(18)

(x, y*)~(x, y*)

with

Ily* I :S c + f + 1 'I]

.

2. Convex Analysis on Banach Spaces

This means that y* !!.. y* in the (j (X* ,X)-topology. Since l.s.c. in this topology, we get from formula (8), since E !!.. 0

Fi (y*) + F; (x*

(19)

Fi

89

and F; are

- y*) ~ £ .

But from the definition of the inf-convolution, we have:

(20)

(FiDF;) (x*)

~

Fi (y*)

+ F; (x*

- y*) ~ £ .

This shows that FiDF; is l.s.c. Formula (4) follows from taking for Fthe ultrafilter consisting of all A c X* with x* E A (which amounts to fixing x* at x* in the preceding argument). 0 This immediately settles the question of finding the subdifferential of a sum: Corollary 4. With the same assumptions, we have:

(21) Proof. In the preceding section, we saw that the left-hand side contains the right-hand side. We have to prove the converse, that is, given any x* E 8(FI + F2 ) (x), to split it into xi E 8FI (x) and x2 E 8F2 (x). Start from x* E 8 (FI + F 2 ) (x). By definition:

(22) Replace (FI

+ F 2 )* (x*)

by its value, given by formula (4):

(23)

x* = xi

(24)

+ X2

.

Rewrite Eq. (23) as follows:

[FI(X)

+ Fi (xi) -

(x, xi)]

+ [F2 (x) + F; (X2)

- (x, X2)] =

0.

Each of the bracket is non-negative, by Fenchel's inequality. Since their sum is zero, both must be zero: (25)

FI(X) +Fi (xi) - (x,xi) = 0

(26)

F 2 (x)

+ F; (x2) -

(x,x2) = 0 .

But this means precisely that xi E 8FI (XI) and x2 E 8F2(X2), and Eq. (24) is the desired splitting. 0 We have similar results for

f 0 A:

Proposition 5. Let X and Y be Banach spaces, A : dom A --+ Y a closed linear operator from X to Y, and G : Y --+ 1R U {+oo} a convex l.s.c. function. Assume that:

90

II. Convex Hamiltonian Systems

°E

(27)

int (domG - A (domA))

Then:

(G 0 A)* (x*) = min {G*(y*) IA*y* = x*} .

(28)

The minimum on the right-hand side is +00 if (A*)-l (x*) = achieved at some point 11* E (A*)-l (x*) otherwise.

0, and is 0

Proof. Define maps G A and G on X x Y by:

GA (X,y ) = {

(29)

G(y)

if y = Ax

+00

otherwise

G(x,y) = G(y) .

(30)

We have domG = x x domG, so that assumption (27) is equivalent to:

°

E int (dom G + graphA) .

(31)

Clearly G A = G + OgraphA. Applying Theorem 3, we get

G*A = G*':* DUgraphA·

(32)

An easy computation gives:

G* (x*, y*) = G* (y*)

(33) (34)

O;raphA

+ Oo(x*)

= O(graphA).L = OgraphA*

•

We write this information back into formula (32), together with formula (4). We get:

G'A (x*, y*) = min {G* (y~)1 y~

°

+ y~ = y*, + X2 = x*, -A*y~ = X2}

Hence:

(35)

G'A(x*,O) = min{G*(y*)IA*y* =x*} . On the other hand, by the definition of Fenchel conjugation, we have:

G'A (x*, 0) = sup {(x,x*)

(36)

x,Y

= sup {(x, x

+ (y,O) -

GA(x,y)}

x*) - G(Ax)} = (G 0 A)* (x*)

Comparing formulas (35) and (36), we get our result.

o

Corollary 6. With the same assumption, we have:

(37)

(G 0 A) (x) = A*aG(Ax)

"Ix EX.

o

2. Convex Analysis on Banach Spaces

91

Proof. Start from x* E 8(G 0 A)(x). By definition:

(38)

G(Ax)

+ (G 0

A)*(x*) = (x, x*) .

By Proposition 5, there is some y* such that:

(39)

(G 0 A)*(x*) = G*(y*)

(40)

A*y* = x* .

Hence:

(41)

G(Ax)

+ G*(y*) =

(x, A*y*) = (Ax, y*) .

This means precisely that y* E 8G(Ax). Equation (40) then yields x* E A*8G(Ax), and the result follows. 0 If for instance we take G = 80 , the indicator function of {O}, we find G* == O. Condition (27) then means that A(domA) = Y. Proposition 5 then becomes: if a closed operator A is onto, then A * has closed range. Finally, if X is a Banach space, we can talk about differentiable functions on X. There are several possible definitions, the two most important being the following: Definition 7. A function F : A -> lRU{ +oo}, which is finite in a neighbourhood of x, is Gateaux-differentiable at x if there exists some x* E X* such that:

(42)

'VyEX,

. F(x+hy)-F(x) _ (* ) 11m h - x ,y .

h-+O

o

We then write F' (x) := x*.

Definition 8. A function F : X -+ lRU{ +oo}, which is finite in a neighbourhood of x, is Frechet-differentiable at x if there exists some x* E X* such that:

(43)

lim F(x +y) - F(x) - (x*,x) = O. y-+O

We then write F' (x) := x*.

Ilyll

o

Clearly Frechet-differentiability implies Gateaux-differentiability. Conversely, if F is Gateaux-differentiable on an open set n c X and F' : n -> X* is continuous, then F is Frechet-differentiable on n: we then say that F is a C 1 function on n. When F is l.s.c., convex and finite on some neighbourhood of x EX, it must be continuous at x, and hence sub differentiable at x, but it need not be differentiable in any sense. So sub differentiability is a strict extension of differentiability, as the following result shows:

92

II. Convex Hamiltonian Systems

Proposition 9. If F : X -+ lR U {+oo} is convex, l.s. c. and if its subdifferential at x is a singleton, 8F(x) = {x*}, then F is Gateaux-differentiable at x and x* = F'(x). If F : X -+ lRU{ +oo} is Gateaux-differentiable and continuous at x, then 8F(x) = {F'(x)}. 0 The first part of the proposition immediately reduces to a one-dimensional problem, and the second part is another consequence of the Hahn-Banach separation theorems. If we go one step further and look into C 2 functions, we get the following result: Proposition 10. Let X be a Banach space and F : X Assume that its Hessian is positive definite everywhere: (44)

"Ix EX,

(F"(x)y,y) > 0

-+

lR a C 2 function.

Vy E Y/{O}

so that F is strictly convex. Then, dom F* has non-empty interior in X*, and F* is C 2 on int (domF*). We have:

(45)

F' x = x* =? ( )

{

X

= [F*]' (x*)

Ix = [F*]" (x*)F"(x)

o

Proof. Fix x E X, and define x* := F'(x). Consider the equation:

(46)

y*

= F'(y)

.

F' is a C 1 map from X to X*. Since F"(x) E .c (X, X*) is positive definite, it is invertible and we may apply the inverse function theorem, to the effect that Eq. (46) can be solved uniquely for y close to x and y* close to x*. Every such y* belongs to dom F*, which therefore has non-empty interior. Inverting Eq. (46) by the Legendre reciprocity formula yields y E 8F*(y*). By local uniqueness, this must coincide with the C 1 solution we found by the inverse function theorem. So 8F*(y*) must be a singleton, 8F*(y*) = {[F*]' (y*)} where [F*]' is CIon a neighbourhood of x*. The Legendre reciprocity formula now reads: y* = F'(y) -F? Y = [F*]' (y*) .

Differentiating the right-hand side with respect to y: Ix = [F*]" (F'(y))

0

F"(y) .

o

3. Integral Functionals on

r"-

93

3. Integral Functionals on La. Let and

n

be some borelian subset of IRn , endowed with the Lebesque measure, --+ IR U {+oo} some non-negative Borel function:

f : n x IRN

(1) Take 1 :::; a :::; 00 and consider the space LD: LD:. For every x E LD:, define F(x) by:

(2)

F(x)

:=

1

(n; IRN ), henceforth shortened to

f (w, x(w)) dw .

We have thus defined a function F : LD: --+ IR U { +00 }. In this section, we shall collect its main properties, both to illustrate the preceding sections by examples, and to serve for future reference. Proposition 1. Assume that

(3)

Vw

f

is non-negative and l.s.c. with respect to

En,

f(w,')

~.

is l.s.c. on IRN .

Then F is l.s.c. on LD:. Proof. If Xn is a sequence converging to x in LD:, we may extract a subsequence such that:

X n ',

lim inf F(xn) =

(4)

n-too

lim F(x n,)

n'---too

and from the Xn' we may extract another subsequence Xn" which converges almost everywhere. We get: lim inf F(xn) =

(5)

n---tOC:>

(6)

=

Since

(7)

lim F (xnll)

n"--+oo

lim

n"---too

J{f(w,Xnll(w))dw. [}

f is non-negative, we may use Fatou's lemma: ;::: {lim inf f(w,Xnll(w))dw.

if}

n'---too

We now use the pointwise convergence Xnll(W) semi-continuity of f(w, .):

(8)

;::: If(w,X(w))dw

=

--+

x(w) and the lower

F(x) .

We have proved that lim infn-----+oo F(xn) ;::: F(x); that is, F is l.s.c. Clearly, if f is convex with respect to

(9)

Vw E

n

~:

f(w,·) is convex

0

94

II. Convex Hamiltonian Systems

then F will be a convex function on La. If (1), (3) and (9) are satisfied, then F will be a l.s.c. convex function, and one should then try to compute F* and {)F in terms of f* and {) I. We use the obvious notations:

(10)

f*(w; ()

(ll)

{)I(w;~) =

I(w, .)*()

=

{)I(w,

.)(~)

.

Henceforth we use the duality pairing (La, Lf3), with Theorem 2. Assume that n has finite measure, that and that there exists x E V XJ such that:

fa

(12)

Then:

F* (x*) =

(13)

I

a-I

+ {3-1 = l.

satisfies (1), (3), (9),

I (w, x(w)) dw < +00 .

fa f*

(w; x*(w)) dw Vx*

E Lf3 .

o

Proof. Introduce the sequence In defined by:

In(w,~) := I(w,~) + ~n 11~lla

(14)

We define accordingly

(15)

In

We note that, for x* E Lf3, the integrals f* (W; x* (w)) dw and I~ (w;x*(w)) are well-defined, with values in lR U {+oo}. Indeed, byassumption (12) we have:

In

(16)

1 n

f* (w;x*(w)) 2: I~ (w;x*(w)) 2: (x(w),x*(w)) - I (w,x(w)) - -llx(w)r

and the right-hand side is an integrable function. The proof now goes in two steps. We first prove the result for the In, and then we take limits. For each fixed n E IN and x* E Lf3, we define Xn (w) by:

(17)

I~

(w,x*(w))

=

(xn(w),x*(w)) - In (w,xn(w))

In other words, xn(w) is the point where the function ~ ---+ (w, x*(w)) attains its maximum. It is well-defined, since In is strictly convex and In(~) 2: ~ II~r· From the latter inequality and the fact that I (w, x(w)) ::; cit also follows that Ilxn(w)11 is bounded independently of w. Hence In(w,~)

(18) and L OO

Xn

c

La since

n has finite

E V XJ

(n; lRN)

measure.

3. Integral FUnctionals on LO:

95

Using the definitions, we have: F~(x*) =

Sup {(x,x*) - Fn(x)}

= Sup [

fa

xEL"

(19)

~

In

((x(w), x*(w)) - fn (w, x(w))} dw

((xn(w), x*(w)) - fn (w, xn(w))} dw

= [fn(W,x*(w))dw= [Sup {(~,x*(W))-fn(w,~)}dw. Jn Jn ~EIRN Now the last term in this string is clearly larger than the first ones. It follows that the inequality is in fact an equality, and formula (13) holds for Fn. We now take limits. Clearly:

\:Ix

(20)

VJI.,

E

F(x) = Inf Fn(x) . n

Dually, we have:

(21)

\:Ix*

E

L{3,

F*(x*)

= Sup F~(x*) n

Indeed, the inequality F :S Fn becomes F*

~ F~

.

by duality. The function

:= Sup n F~ is convex and l.s.c. If iP* i= F*, then there would be a convex function iP := iP** such that iP i= F and F :S iP :S Fn for all n, thereby

iP

contradicting (20). Replacing F~ by its value, we get:

F*(x*) = Sup [

(22)

n

Now, for each fixed w

(23)

f(w,·)

E Q,

In

f~ (w;x*(w))dw.

we have

Inf fn(w,')

and fn+!:S fn

= Sup f~(w,·)

and f~+l ~ f~

=

n

and hence, as we just saw:

(24)

j*(w,·) Since

J

(25)

J; (w;x*(w))dw >

.

-00,

we may apply Lebesgue's monotone convergence theorem. We get: (26)

[j*(w;x*(w))dw=Sup

In

n

[f~(w;x*(w))dw.

In

Formula (13) now follows from (22) and (26).

D

96

II. Convex Hamiltonian Systems

The non-negativity condition (1) may seem stringent. It can easily be weakened. Theorem 1 will still hold if it is replaced by the following:

(27)

[Ir (w;x*(w))dw


Lf3 is C l

(45)

,

{a, cIlxllCl.-l }

and

[F'(x)] (w)

= I' (w,x(w))

o

Proof. First apply Corollary 3. We get

(46)

8F(x)

=

{x* E Lf3lx*(w)

= I' (w, x(w))

a.e.}

By inequality (44), whenever x E LCI., then I' (·,x(·)) E Lf3. So the right-hand side of formula (46) is a singleton, and F is Gateaux-differentiable everywhere by Proposition 2.9, with F'(x) given by formula (45). It then follows from Theorem 4 that F' : LCI. -> Lf3 is continuous. 0 Notes and Comments. Krasnoselskfi's theorem (to be found in [Kra], together with its converse) does not hold for maps into L oo • This has been the source of many mistakes. For instance, if a map F : L2 -> lR defined by the formula

(47)

F(x) :=

In

f(w, x(w) dw

happens to be C 2 , then it must be exactly quadratic, that is, we must have f(w,x) = (A(w)x,x) for every (w,x) (see [Skrl] and [Skr2]).

4. The Clarke Duality Formula Let X be a Banach space and F : X -> lR U { +oo} a convex l.s.c. function on X. We are given a closed linear operator A from X to X*. Recall that this means that A maps a linear subspace dom A c X into X* , and that graphA c X x X* is closed. We assume that A is self-adjoint:

4. The Clarke Duality Formula

(1)

(x,Ay) We define tJ> : dom A

(2)

= (Ax,y) V(x,y) -+

E

99

(domA)2

1R{ +oo} by:

tJ>(x)

:=

1

"2 (x, Ax) + F(x) .

tJ> is the sum of two terms: a quadratic form and a convex l.s.c. function. The quadratic form is not required to be non-negative definite, so tJ> need not be a convex function. Indeed, in the cases which shall be of interest to us, tJ> will turn out to be quite complicated; for instance, it will have infinitely many critical points. A critical point of tJ> is defined to be a point x E dom A where: (3)

Ax+oF(x) '" 0

and tJ>(x) is then called a critical value. If tJ> is C l , these are standard definitions. If tJ> is not smooth, a little work may be needed, as in: Proposition 1. Assume that the restriction of tJ> to any straight line running

through x has a local minimum or a local maximum at x. Then x is a critical point of tJ>. Proof. Assume that x is not a critical point, so that -Ax does not belong to of(x). This means that there exists some point Xl E X and some E > 0 such that: F(xd :s; F(x) - (Ax, Xl - x) -

(4)

E •

Set Xt := tXl + (1- t)x. The function t -+ F (Xt) defined on the real line is convex, so that its slope [F (Xt) - F(x)] It is a decreasing function of t. Hence, for all t E (0,1):

(5)

1

t

[F (Xt) - F(x)] :s; F (Xt) - F(x) :s; -(Ax, Xl - x) -

E •

Write a := (A (Xl - x), Xl - x) so that 1

1

t2

"2( Ax t,Xt) = "2 (Ax, x) +t(AX,Xl-X) +a"2 .

(6)

Combining the last two relations, we get for 0 < t < 1

(7)

1

1

t2

F(x) + "2 (Axt, Xt) :s; F(x) + "2 (Ax, x) - Et + a"2

and for t < 0

(8) If we choose It I :s; 2Elal- l , then -Et+a~ has the same sign as -Et, negative in the first case and positive in the second. This proves that

100

II. Convex Hamiltonian Systems

(9)

gi(Xt)

< gi(x)

for 0

< t < 2Elal-1

(10)

gi(Xt)

> gi(x)

for - 2Elal- 1 < t

O:

X

0

1R U {+oo}, with

is convex l.s.c. on 1R2n

(53) (54)

[0, T]

is Borelian on

----+

= M.

H (t, x(t)) dt
0 and the symplectic matrix M are prescribed. We associate with it the operator A defined by formula (33) and the functional ip : dom A ----+ IR U { +oo} given by:

ip(x)

1 = -(Ax,

2

(58) =

x)

iT D

+

iT a

(Jx,x)

H(t,x)dt

+ H(t,x)]

dt.

4. The Clarke Duality Formula

105

Proposition 5 (Least action principle). The critical points of cJj are exactly the solutions of problem (57). 0 Proof. For x E La, set 1i(x) := JoT H (t, x(t)) dt. It is a convex l.s.c. function. The assumption of Theorem 4.2 and Corollary 4.3 are satisfied, with (1) replaced by (27), so that

(59)

1i*(x*) =

(60)

81i(x)

iT

=

H* (tjx*(t))dt

Vx* E L(3

{x* E L(3lx* E 8H(t,x(t)) a.e.}

By definition, x is a critical point of cJj if and only if Ax E 81i(x), that is:

± E L(3

(61)

J± + 8H (t, x(t)) ::l 0

and

a.e.

Hence ± E J8H(t,x), as desired. 0 We now wish to apply Theorem 2. Introduce a new functional tf/ on La: tf/(x)

= ~(Ax,x) +1i*(Ax)

(62) =

iT [~(JX,

x)

+ H*(t, -Jx)]

dt .

Proposition 6 (Dual action principle). x is a critical point of tf/ if and only if there is some constant ~ E lR2n such that x(t) + ~ solves problem (57). 0 Proof. Defining L~ by formula (44), we have:

(63)

L~

c

A(domA)

c

L(3 .

By condition (54), denoting by B (lR2n) the unit ball, we have:

EB (lR2n) C dom 1i* .

(64) Hence:

(65) Condition (14) is satisfied, and we may apply Theorem 2. The result 0 follows. If M = I, and H is T-periodic (H(t + T,x) = H(t,x) for all t and x), or autonomous (H(x) independent of t), problem (57) becomes

(66)

{

± E JH(t,x) x(O)

= x(T)

and x is now a T-periodic solution. We shall dwell a length on this particular case in subsequent chapters. For the time being, we conclude this section with a few more examples, which we treat in lesser detail.

106

II. Convex Hamiltonian Systems

Example 1: The Bolza Problem We are still dealing with the same equation x E J{)H(t, x), where the Hamiltonian H satisfies conditions (51) to (54), but this time we consider a fixedendpoints problem. Split x into (p, q), with p and q E lRn. Take two points qo, ql E lRn and some time T > O. We are interested in the boundary value problem:

!

~: H;(;,P) p - -Hq(q,p)

(67)

q(O) = qo

q(T)

=

ql .

Introduce the function q(t) := (ql - qo) tiT, and the operator A : U" Lf3 defined by: {

(68)

domA: {(P: q~lp E Lf3, q E Lf3, q(O)

---+

= 0 = q(T)}

A(p, q) - (-q,p) .

Now define the action functional iP on the linear space La. x dom A by:

(69)

iP(q,p):=

i

T

[-p(q+:)+H(p,q+q)]dt.

If (p, q) is a critical point of iP, then (p, q+q) is a solution of problem (67), and conversely. Now apply Theorem 2. We write iP as

iP(x)

(70)

=

1

2(Ax,x)

+ 1t(x)

where A is defined by formula (68) and 1t by:

(71)

1t(p,q) =

iT

[-P: +H(P,q+q)] dt.

The operator A is closed and self-adjoint. The conjugate 1t* is given by:

(72) The dual functional If! given by Theorem 2 now is:

If!(p, q) : = (73)

iT

[-pqpq + H (q + : ' -p) ] dt

4. The Clarke Duality Formula

107

The kernel of A consists ofthe constant functions ((,0), with (E lRn. So (p, q) is a critical point of lJ! if and only if there is some constant ( E lRn such that (( + p, q + q) solves problem (67). Of course, in the above formulas, !f!f should be replaced by the constant

(ql - qo)/T.

Example 2: Lagrangian Formulation

In classical mechanics, the Hamiltonian splits in two parts, kinetic and potential energy. Typically, we have:

(74)

H(t,p, q) =

1

2 (A(t)p, q) + V(t, q) ,

where A(t) = A(t)* is a positive definite matrix, depending continuously on t, and V : [0, T] x lRn ---7lRU {+oo}, the potential, is measurable in t and convex l.s.c. in q. The Hamiltonian system x = J H' (t, x) becomes a second-order system for q in lRn :

!

(75)

(A(t)q)

+ 8V(t, q) =

0

with p = A(t)q. To study Eq. (75) it is better to use the Lagrangian formalism. Instead of the Hamiltonian action, we shall use the functional: (76)

p(q)

iT [-~

=

(A(t)q,q)

+ V(t,q)] dt.

Assumptions (51) to (54) on the Hamiltonian translate into assumptions on the potential v:

"it

(77)

:3q

(78)

(79)

E [0, T],

:310

> 0,

E

((

VXJ,

E

V(t,·) is convex l.s.c.

iT

lRn , 11(11:::;

V (t, q(t)) dt
O. Similarly, the Hamiltonian

112

III. Fixed-Period Problems: The Sublinear Case

(6)

1 H(t,x) = "2k

Ilxll 2 + Ilxl let

is (k, k + €)-subquadratic for every € > O. Note that, if k < 0, it is not convex. Suppose H is (Aoo, Boo)-subquadratic at infinity. We are interested in solving the boundary-value problem: {

(7)

X E JAoo(t)x

+ J8N(t,x)

a.e.

x(O) = x(T) .

If His C 1 and T-periodic with respect to t, for instance, if the system is autonomous, this amounts to finding T-periodic solutions of the Hamiltonian system:

x=

(8)

JH'(t, x) .

Arguing as in Sect. 4 of the preceding chapter, we find that the solutions of problem (8) are the critical points of the action functional:

(x)

=

iT [~(JX

+ Aoo(t)x, x) + N(t, x)] dt

on the space W 1 ,2 (IR/TZ; IR2n) of T-periodic absolutely continuous functions with square-integrable derivative. Introduce the operator A on L2 defined by:

(9) (10)

domA = W 1 ,2 (IR/TZ;IR2n) d

A=J dt +Aoo. Its main properties are summarized in the following:

Proposition 2. The operator A is closed and self-adjoint. Its range R(A) := A(domA) is closed in L2, its kernel Ker A is 2n-dimensional at most, and L2 splits orthogonally:

(11)

L2 = Ker A EB R(A) .

Endow dom A with its standard (graph) norm and R(A) with the L2-norm, and consider the double restriction Ao of A:

(12)

Ao : domA

n R(A)

-+

R(A) .

Then Ao is a Hilbert space isomorphism, and its inverse A~l is compact. D

Before proceeding with the proof, note that if Ker A and A : W 1 ,2 -+ L2 is invertible with compact inverse.

= {O}, then A = A o ,

1. Subquadratic Hamiltonians

113

Proof. All these are well-known properties of the differential operator A == J-ft + Aoo(t) over T-periodic functions. To invert A, we have to solve the equation

(13) which has the explicit solution (variation of constants formula)

(14)

x(t) == R(t)x(O)

-it

R(s)-lJu(s)ds .

Here R(t) is the matrizant

{

(15)

-ftR(t) == JAoo(t)R(t) R(O) == I .

Throwing in the boundary condition x(T) == x(O), we get a relation between x(O) E lR2n and u E L2 (16)

(R(T) - I) x(O) ==

iT

R(t)-l Ju(t)dt .

This determines x(O) in termes of u, provided R(T) does not have the eigenvalue 1: the operator A then is invertible. If R(T) has the eigenvalue 1, the right-hand side u of Eq. (16) must satisfy the relation

(17)

iT

R(t)-l Ju(t)dt E (R(T) - I) (lR2n)

which characterizes the range of A. Its codimension in L2 must therefore be less than or equal to 2n. We now check that the range of A is the orthogonal of its kernel. Take any v E Ker A. We have:

(18)

v(t) ==

R(t)~

,

with

~ E

Ker (R(T) - 1)

and:

(19)

(U,V,)£2 == iT (u(t),R(t)~)dt== iT (R*(t)u(t),~)dt. Remember that R(t) is symplectic:

114

III. Fixed-Period Problems: The Sublinear Case So u E L2 is orthogonal to Ker A if and only if:

iT R(t)-l Ju dt

(21)

E [J Ker (R(T) -

I))-i

Since R(T) is symplectic, we have the identity (R(T)C - (, J~) = ((, R(T)* J~ - J~) = (J(, ~ - R(T)-l~)

(22)

from which it follows that the right-hand sides of Eqs. (17) and (21) coincide. So u E (Ker A)-i if and only if u E R(A), as announced. From the orthogonal splitting

L2 = Ker A EEl R(A)

(23)

it follows that A o is one-to-one and continuous. By the closed graph theorem, it must be an isomorphism. Denote by Xu E W 1 ,2 the preimage A;;-lU. Since A;;-l is continuous, and W 1 ,2 is contained in Co, the map u --t xu(O) from R(A) into IR2n is continuous. It then follows from the integral representation (14) that the map u --t Xu = A;;-lu is compact. 0 Now define a convex l.s.c. map ./1/ on L2 by:

N(X) iT N (t, x(t)) dt .

(24)

=

Set:

(26) By assumption (4), n*(p) is finite everywhere. We have: (27)

from which it follows that: (28)

N"(y) =

(29)

8N"(y) =

iT N* (t; y(t)) dt {Z

E

L21 z(t)

E 8N* (t,y(t))

a.e.}

(Theorem 11.3.2 and Corollary II.3.3). We find that the domain of N" contains Loo. Since the kernel of A consists of C 1 functions, it is contained in L oo , and condition (14) of Theorem 11.4.2 is satisfied. It then follows from Theorem II.4.2 that solutions of problem (7) are related to critical points of the dual functional

1. Sub quadratic Hamiltonians

w(x) : =

(30) =

115

~ (Ax, x)£2 + N*( -Ax)

iT [~(JX

+ AcXl(t)x,x) + N*

(t, -Jx - ACXl(t)X)] dt .

Here x E dom A = W l ,2 (lllITZ; lll2n). In accordance with Proposition 2, it will be convenient to make the change of variables Ax = u, thereby introducing the reduced functional

(31)

't/J: R(A) -+ lllu {+oo}

defined by: 't/J(u) : =

(32) =

~ (A~lU,u)£2 +N"(-u)

iT [~(A~lU,U) +

N*(t, -u)] dt .

The results of Theorem 11.4.2 are summarized in the following: Theorem 3. ffu E R(A) is a critical point of't/J, there will exist some Xo E Ker A such that

(33) solves problem (7). Conversely, if x E W l ,2 solves problem (7), then u := Ax is a critical point of't/J on R(A). 0

The question now is to find critical points of 't/J on dom A. Clearly, the spectrum of Ao will play an important role. By construction:

(34)

Spec Ao

= (Spec A) \ {O} .

Proposition 4. The spectrum of Ao consists of a doubly infinite sequence An, n E Z of reals:

(35) (36) (37)

An ::::; An+l An -+ +00 An -+ -00

'in as n as n

-+ +00 -+ -00 .

Each eigenvalue An has multiplicity at most 2n.

o

Proof. Since A~l is compact and self-adjoint, its spectrum consists of a sequence iLk, k E :IN, of reals, with IiLk I -+ o. The spectrum of Ao consists of the sequence Ak := iLk l so that IAkl-+ 00. We now check that the spectrum of Ao contains arbitrarily large positive and negative terms. If this were not the case, it would be bounded from above or from below: assume for instance it is bounded from below:

III. Fixed-Period Problems: The Sublinear Case

116

(38) Then Ao -

X would be positive semi-definite:

(39) If x E dom A, we can write x = Xo + y, with Xo E dom Ao and y E Ker Ao. So the above relation immediately extends to

(40)

(AX,X)£2 -

XllxIIL2

2:: 0

Vx E domA

.

Now define a test function xp as follows:

(41) with p E 7l.. and ~ E is orthogonal:

lR,2n

with II~II = 1. We have, since the operator

eP21fJt/T

(42) (Axp, xp)£2 = -27rp +

(43)

iT

(AcXl(t)XP(t), Xp(t)) dt

2:: -27rp + a~T ,

where

a;t, is a constant, chosen in such a way that

(44) for 0 ::;: t ::;: T and 11(11 = 1. This is possible by compactness. If we choose p > (a;t, - X) /27r, inequality (43) contradicts (39). So condition (38) cannot hold. We have proved that the spectrum of Ao is unbounded on both sides. D Proposition 4 now is a matter of ordering the Ak. For instance, if Aoo = 0, we have:

(45)

A=J~ dt

~ {u E L' (O,T;IR'") liT udt ~ o}

(46)

RCA)

(47)

A;'u~x [u~J±

and

iT Xdt~ol.

In other words, A;;-lu is just the primitive of u with mean value zero. In this particular case, the spectrum can be computed explicitly:

2. An Existence Result

117

(48) (49) Each eigenvalue has multiplicity 2n. If A = corresponding eigenspace is

2,; k

is an eigenvalue, the

(50)

2. An Existence Result In this section, we consider a Hamiltonian

(1)

H(t, x) =

21 (Aoo(t)x, x) + N(t, x)

which is (Aoo, Boo)-subquadratic at infinity, and we want to solve the boundaryvalue problem:

{

(2)

i: E JAoo(t)x

+ J8N(t,x)

x(O) = x(T) .

We recall that the time-dependent matrix:

(3) is assumed to be positive semi-definite. Denote by 0 :S Il(t) :S ... :S 12n(t) its eigenvalues, and set:

(4) Recall also that the eigenvalues of Ao form an increasing sequence An, nEll., with An - 7 ±oo as n - 7 ±oo, and An -I- 0 for all n. Denote by A the largest negative eigenvalue:

A := Max {An < O} .

(5) Theorem 1. If

IAI > I,

then problem (2) has one solution at least.

D

Before proving Theorem 1, let us give some consequences in various settings: first, the case when Aoo = 0; then the case when Coo = 0; finally a case when the assumptions are given in terms of the Hessian H"(t, x).

Proposition 2. Let H(t, x) be a Hamiltonian, Borelian with respect to all variables, convex with respect to x for fixed t, and such that:

(6)

p-1Min {H(t,x)IO:S t:S T,

Ilxll

= p}

-7

+00

III. Fixed-Period Problems: The Sublinear Case

118

(7)

1

for some

boo > 0

and

c E JR.

2

2boo Ilxll + c V(t, x)

H(t, x) ::;

Then, provided:

(8) the problem:

(9)

± E J8H(t,x)

(10)

x(O) = x(T)

o

has a solution.

it

Proof. Here Aoo = 0, so A = J and the spectrum of Ao is pointed out at the end of the preceding section. So

A=

(11)

Coo

On the other hand, Boo = from Theorem 1.

=

_

2,; 7l. \

{O}, as we

211'" T·

boo!,

so 'Y =

boo.

The result now follows 0

Proposition 3. Assume the Hamiltonian H(t, x) is given as: (12)

H(t,x) =

1

2 (Aoo(t)x,x) + N(t,x)

with Aoo(t) = A~(t), continuous with respect to t, and N(t, x), Borelian in (t, x) such that

(13)

is convex in x for fixed t

N(t, .)

(14)

N(t, x) ~ n (1Ixll)

(15)

N(t, x) ::;

with n(s)s-l

m(1Ixll)

--7 +00

with m(s)s-2--70

as s as

--7 00

S--7OO.

Then the problem:

(16)

{

± E J Aoo(t)x + 8N(t, x) x(O)

= x(T)

o

has a solution for every T > O. Proof. Take any T > O. Condition (15) means that, for every able to find some constant c E JR such that: (17)

N(t, x) ::;

E

2

211xll + c .

E

> 0, we will be

2. An Existence Result

119

From this it follows that H (t, x) is (A>e' Aoo + €I)-subquadratic at infinity for every 10 > O. In other words, we may take Coo = €I, and hence 'Y = 10. Taking 10 < A, we are in a position to apply Theorem 1. 0 Proposition 4. Let the Hamiltonian H(t, x) be C 2 . Assume that two real numbers a and b can be found with

(18)

for some ko E 71.

(19)

aI:::; H"(t,x) :::; bI i/(t,x). Then the boundary-value problem {

(20)

X = JH'(t,x)

x(O)

= x(T) o

has one solution at least.

Proof. It follows from the mean value theorem that H(t, x) is (a, b')-subquadratic at infinity for every b' > b. We may then take 'Y = b' - a in Theorem 1. On the other hand, A = J + a, so

-it

(21)

Spec A

The largest negative eigenvalue A is by:

27l"

= a + ""T71. . (a +

2; kI)

where ki E 71. is defined

(22) The condition

(23)

IAI > 'Y of Theorem 1 now becomes 27l" , --ki - a > b - a T

which amounts to: (24) Setting ko := -(ki + 1), we find that formulas (22) and (24) reduce to condition (18). Hence the result. 0 We now proceed to the proof of Theorem 1. We shall show that the dual action functional:

(25) attains its minimum on R(A), and the result will then follow from Theorem 1.3.

120

III. Fixed-Period Problems: The Sublinear Case

Recall first that A;;-1 is a compact self-adjoint operator from R(A) into itself, where R(A) c L2 is a closed subspace with the induced topology. Its eigenvalues are the >.;;1. Defining>. > 0 by (5), we get

(26) On the other hand, we have, since H (t, x) is (A"", Boo )-subquadratic at infinity:

(27)

N(t,x)

1

2 (Coo(t)x,x)+c V(t,x).

~

Here Coo(t) = Boo(t) - Aoo(t) is positive semi-definite. Remembering that N (t, x) is superlinear with respect to x:

(28)

n (1Ixll)

~

N(t, x)

~

1

2 (Coo (t)x, x) + c ,

with n(s)s-1 --+ +00 as s --+ 00, we see that Coo(t) is in fact positive definite, and hence invertible. Taking conjugates in inequality (28) then yields:

(29)

N*(t;x) ~ So the number

(30)

1

2 (Coo (t)-1 x ,x) -

'Y defined by (4) N*(t,x)

V(t,x).

C

is in fact positive, and we have: 1

~ 2'Y-

1

Ilxll 2 -

c.

Writing inequalities (26) and (30) into the functional 'IjJ, we get: (31)

'IjJ(u) ~

1

2 (>.-1 + 'Y- 1 ) IIuII 2 -

c.

Since I'YI > >., we have 'Y- 1+>.-1 > 0, so that 0 such that, for every T 'IjJ yields a non-constant T -periodic solution x.

> To, minimization of D

Proof. The assumption implies that H is (a oo , b)-sub quadratic at infinity for every b > a oo . It is easily checked that aoo < - E [aooJ. As soon as:

i:r

i:r

3. Autonomous Systems

(26)

a

125

[T ]

211"E - - a < b< - T 211" =

=

the left inequality in (18) will be satisfied. On the other hand, the right inequality in the same formula can be rewritten as:

(27) As T ----> 00, the left-hand side goes to a=, while the right-hand side w is the lowest eigenvalue of a= + N"(~), when ~ is an equilibrium. It then follows from assumption (25) that a= < w, so that inequality (18) is satisfied for large T. D For instance, if a= = 0, that is if H(x) is convex C 2 and such that

(28) (29)

H(x)

Ilxll- 2----> 0

as

H'(x) = 0 =:} H"(x) 2:: wI

Ilxll----> 00 with w> 0

Erw

then we find a non-constant T-periodic solution whenever > 1, that is T > ::. Note that is an upper bound for the periods of the solutions to the linearized system at equilibrium, iJ = JH"(~)y. In fact, in this example, x has minimal period T (which implies nonconstancy), as we now show.

2:

B. Minimal Period Minimizing the dual action functional 1lf(x) over domA (or equivalently, minimizing 1lf(u) over R(A)) yields a T-periodic solution x of x = JH'(x):

(30)

1lf(x) :::; 1lf(x) \:Ix E domA .

We would like to know what the actual period of x is. It is T-periodic, to be sure, but does it have a lower period: T /2, perhaps T /10 6 ? In other words, it is just the k-th iterate of some T /k-periodic solution? If such is not the case, that is if x is never T /k-periodic for any integer k 2:: 2, we shall say that x has minimal period T. To simplify matters, we shall assume that His (0, b=)-subquadratic, that is: H : IR2n

(31)

----> IR Ilxll- 1 ----> +00 H(x) :::; !b= IIxl12 + C

{ H(x)

is convex, when

and that it attains its minimum at the origin

Ilxll ----> 00

126

III. Fixed-Period Problems: The Sublinear Case

{

(32)

H(x) > H(O) = 0

\:Ix i= 0

lim inf2H(x) Ilxll- 2

:=

IIxll-+O

8> 0 .

Note that H* also attains its minimum at the origin, and that this minimum is zero:

(33)

H*(O) = Max ((O,x) - H(x)} = 0

(34)

o E oH(O) ~ 0 E oH*(O)

x

by the Legendre reciprocity formula. The dual action functional then is: (35)

to be minimized on the space:

(36)

W 1 ,2 (lRITZ; lR2n) = {xix E L2

-it

(0, T; lR2n) ,x(O) = x(T)}

e

The operator A = J has a kernel, namely the constants E lR 2n. Any minimizer oflP is of the form x(t)+e, with E lR2n and x a T-periodic solution of

e

x = JoH(x) .

(37)

Proposition 4. Assume (31) and (32), together with

(38) Then

x has

o

minimal period T.

Proof. The only equilibrium is the origin O. By Corollary 2, we have:

(39)

IP(x) < IP(O)

so that x ~ lR2n : the solution x is non-constant. We have to show that it has minimal period T. Assume otherwise, so that the minimal period is T Ik for some k ~ 2. Define a new function x k by (40)

Xk(t)

:=

kx(tlk) .

Then Xk is still T-periodic. We may therefore substitute into W, yielding thereby:

3. Autonomous Systems

tP(Xk) = iT (41)

=

iT

127

[~(JXk'Xk)+H*(-JXk)]dt

[~ (JX (~) ,x (~)) + H* (-JX (~))] dt

= k iT/k

[~(J~,x) +H* (-J~)] ds.

Since x is T I k periodic, the right-hand side satisfies:

(42)

But we know that:

and since

H* ;::: 0, we must have: iT ~

(44)

(J ~ ,x) dt < 0 .

Writing this into formula (42) we get

(45) thereby contradicting the fact that

x is a minimizer.

D

This argument uses the fact that x is a global minimizer. One can do better by appealing to index theory. In fact: Lemma 5. Assume that H is C 2 on IR2n \ {O} and that H" (x) is non-degenerate for any x -# O. Then any local minimizer of tP has minimal period T.

x

Proof. We know that x, or x + ~ for some constant ~ E IR2n , is aT-periodic solution of the Hamiltonian system:

x = JH'(x) .

(46)

There is no loss of generality in taking ~ = O. So tP(x) ;::: tP(x) for all x in some neighbourhood of x in W 1 ,2 (IR/TZ; IR2n). Clearly C 1 C W 1 ,2 with a stronger topology. So the restriction of tP to C 1 (IR/TZ;IR2n), which we denote by~, has as a local minimizer. On the other hand, ~ is clearly C 2 on a neighbourhood of X, and

x

(47)

(~" (x)y, y) =

iT [(Jy, y) + ((H*)" ( -J~!) Jy, JY) ] dt

128

III. Fixed-Period Problems: The Sublinear Case

x

for y E C 1 . Since is a local minimizer, this quadratic form is semi-positive definite. We have, by Proposition 11.2.10, remembering that -J~~ = H'(x),

(H*)" ( -J~~ (t)) = H" (X(t))-1 .

(48)

Both sides depend continuously on t. Therefore, C 1 being dense in W 1 ,2, the quadratic form offormula (47) extends to a quadratic form on W 1 ,2 given by:

iT [(Jy,

(49)

y)

+ (H" (X(t))-1 Jy, JY)]

dt

which is also semi-positive definite. In other words, its index is zero. But this index is precisely the index iT(x) of the T-periodic solution x over the interval (0, T), as defined in Sect. 1.6. So

iT(X)

(50) Now if

(51)

= 0 .

x has a lower period, T/k say, we would have, by Corollary 1.6.5: iT(X) = ikT/k(X) 2 kiT/k(X)

+k -

12k - 1 2 1 .

This would contradict (50), and thus cannot happen.

o

For future reference, we spell out on easy consequence of Proposition 4. Corollary 6. Let H : 1R2n

-+

1R be a convex function such that

H(x) 2 H(O)

(52) (53)

H(x)

Ilxll- 2 -+ 00

(54)

H(x)

Ilxll- 2 -+ 0

= 0

when when

Ilxll Ilxll

-+

0

-+ 00 .

Then, for every T > 0, the system (55)

X E JH(x)

(56)

x(O)

has a solution with minimal period T.

=

x(T)

o

Notes and Comments. The results in this section are a refined version of [ClaE3]; the minimality result of Proposition 4 was the first of its kind. To understand the nontriviality conditions, such as the one in formula (20), one may think of a one-parameter family XT, T E (21fw- 1 , 21fb;;}) of periodic solutions, XT(O) = xT(T), with XT going to 0 when T -+ 21fw- 1 , which is the smallest period of the linearized system at O.

4. Nonautonomous Systems

129

Unfortunately, although this picture is correct in the homogeneous case, = Ilxll'" with 1 < a < 2, I have been unable to support it by a priori estimates in the general case. As far as we know, the XT could be bounded away from 0 and infinity when T ranges over (27rw-1, 27rb;;;,l). It would be interesting to settle this question. It would also be interesting to know what happens for T > 27rb~l. The minimization method fails, but one expects that, for most if not all systems, there will be periodic solutions for arbitrarily large minimal periods.

H(x)

4. N onautonomous Systems Consider a nonautonomous Hamiltonian H(t, x), T-periodic in t:

(1)

H(t+T,x) = H(t, x)

V(t, x) E JR

X JR2n

and the associated system:

x=

(2)

JH'(t,x) .

Such systems have a natural period, namely T. Except in special cases (an example of which will be seen presently), there are no constant solutions, so we may apply the existence results of Sect. 2 without worrying about showing that the periodic solutions we obtain are non-trivial. The question of minimality is also much simplified: it will generally be the case that, if T is the minimal period of the Hamiltonian H, it will also be the minimal period of any T-periodic solution. There remains, however, the possibility of subharmonics. Take an integer k, and let us seek a kT-periodic solution Xk of the Hamiltonian system:

(3)

{

Xk

=

JH'(t,Xk(t))

Xk(t + kT) = Xk(t) .

Clearly, if Xl exists, it is T-periodic and hence kT-periodic for all integer = Xl solves problem (3). Any Xk =f Xl for k =f 1 is called a sub harmonic. The question then is whether we can find for each k ~ 2 a subharmonic Xk, such that the Xk are pairwise distinct. No such result is known in the sub quadratic case. Failing that, one may ask whether there exists an infinite sequence Xk, k E A c IN, of pairwise distinct subharmonics. An easy answer is provided by taking A c P, where P denotes the set of all primes. To prove that the Xk, k E A, are pairwise distinct, it will then be enough to prove that Xk =f Xl for all k E A. We give a stronger result, which will imply that non-trivial subharmonics Xk exist also for non-prime k.

k, so that Xk

130

III. Fixed-Period Problems: The Sublinear Case

Proposition 1. Assume H(t,x) is (O,E)-subquadratic at infinity for all and T -periodic in t (4)

H(t,·)

(5) (6)

(7)

H(.,x) H(t,x);:::

n(lIxll)

VE> 0,

E

> 0,

is convex Vt is T-periodic Vx with n(s)s-l -+

3c : H(t,x):::;

00

as s -+

00

E 2 "2l1xll +c.

Assume also that H is C 2 , and H" (t, x) is positive definite everywhere. Then there is a sequence Xk, k E 1N, of kT-periodic solutions of the system

x = JH'(t,x)

(8)

such that, for every k E 1N, there is some Po E 1N with:

(9)

o

Proof. For each integer k, we consider the functionallJtk on W 1 ,2 (lRITZ; lR2n) defined by:

(10) By the results of Sect. 2, the minimization of IJtk is possible for every k, and yields a kT-periodic solution Xk. Arguing as in Proposition 3.4, we see that Xk has index on the interval (0, kT). If Xpk = Xk, then Xk still has index on the interval (O,pkT), since it minimizes IJtpk :

°

°

(11) On the other hand, by Proposition 1.5.20, we have:

(12) This shows that equality (11) ceases to hold when p is large enough. Hence the result. 0 Let us illustrate all this in two simple situations. Example 1 (External forcing). Consider the system:

(13)

x = JH'(x) + f(t)

where the Hamiltonian H is (0, boo)-subquadratic, and the forcing term is a distribution on the circle:

4. Nonautonomous Systems

131

(14) where fa := T- 1

(15)

t' f(t)dt. For instance, f(t) = I: Ok~ , kE71.

where Ok is the Dirac mass at t = k and ~ E IR2n is a constant, fits the prescription. This meanS that the system x = J H' (x) is being excited by a series of identical shocks at interval T. Formally, the Hamiltonian corresponding to Eq. (13) is:

H(t,x):= H(x) - (Jf(t,x),x)

(16)

Hence the dual action functional:

(17)

qi(x) =

IT[~(Jx,x)+H*(-Jx+Jf(t))]dt.

Change variables by setting x = F(t)

(18)

iP(y) =

+ y. We get qi(x) =

iP(y) with:

IT[~(JY+J(f-fo),y+F)+H*(-JY+Jfo)]dt.

Throwing away the term J (J(f - fo),F), which is constant and therefore plays no role in the minimization, and integrating by parts the term J (J(f - fa), y), we are led to the following lemma, which we leave to the reader: Lemma 2. Consider the functional qi on W 1 ,2 (IR/TZ; IR2n) defined by:

If y is a critical point of lif, then there is a constant ~ x = y + F(t) + ~ is a T-periodic solution of system (13).

E

IR2n such that D

As a corollary, we see that if H is (0, b(Xl)-subquadratic at infinity, with (20) then system (13) has a T-periodic solution (see Proposition 2.2). Example 2 (Parametric excitation). Consider the system

(21)

x

=

Ja(t)H(x)

where the Hamiltonian H is (0, E)-subquadratic at infinity for all E > 0, and where a : IR ---t IR is a continuous T-periodic function:

132

III. Fixed-Period Problems: The Sublinear Case

< 0:

a(t) ~ f3

(22)

0

(23)

a(t + T) = a(t) .

~

Assume moreover that H is C 2 and has its unique equilibrium at the origin: (24)

H(x) > H(O)

(25)

H"(x)

= 0

for x

-I- 0

positive definite \Ix

and denote by w the lowest eigenvalue of H"(O):

H"(O) 2: wI .

(26)

So x = 0 is a solution of the autonomous problem x = JH'(x). 1fT < ~, it is to be expected that it is the only one. Clearly, the nonautonomous system (21) also has the trivial solution x = O. We now look for subharmonics:

{

(27)

Xk = Ja(t)H' (Xk) Xk(t + kT) = Xk(t) .

We find Xk by minimizing, for each k E IN, the dual action functionallJlkT. Proposition 1 then applies and tells us, for every k E IN, there is some Po with Xpk -I- Xk for all p 2: Po. In particular, there will be some ko such that Xk -I- 0 for all k 2: k o . Again, if H is sub quadratic, the analysis in the preceding sections can be carried over, with suitable changes in the numerical constants. Here is a typical result. Proposition 2. Assume that:

(28)

H"(x)

is positive definite for all x E IR2n

(29)

H(x) 2: n (11xll)

(30)

H(x)

~

m(1Ixll)

with n(s)s-l

as s

-> 00 .

0 as s

-> 00 .

-> 00

with m(s)s-2

->

.

Then problem (21), with a(t) T-periodic and bounded away from zero and

+00, has an infinite sequence of pairwise distinct T -periodic solutions, the minimal period of which goes to infinity.

0

Notes and Comments. The first results on subharmonics were obtained by Rabinowitz in [Rab3], who showed the existence of infinitely many subharmonics both in the subquadratic and superquadratic case, with suitable growth conditions on H'. Again the duality approach enabled Clarke and Ekeland in [ClaE3] to treat the same problem in the convex-sub quadratic case, with growth conditions on H only.

5. Other Problems

133

Recently, Michalek and Tarantello (see [MicT] and [Tar]) have obtained lower bound on the number of subharmonics of period kT, based on symmetry considerations and on pinching estimates, as in Sect. 5.2 of this book.

5. Other Problems As we mentioned earlier, the duality method will apply to other situations. We mentioned two specific examples in Sect. 11.4, and we pick them up again. A. Second-Order Systems

We consider the boundary-value problem:

{

(1)

ij+V1(t,q) =0

q(O) = q(T) ,

q(O) = q(T) .

According to Example 2 of Sect. HA, the direct functional in the Lagrangian formalism is:

(2) while the dual is:

(3) All the analysis in the preceding sections can be carried over to this case, with different values for the numerical constants, leading to different bounds for T. To be precise, we have, by Wirtinger' inequality:

iT q2dt :::;

(4)

~22iT ij2dt

which will replace the inequality:

iT (J±, x) dt :::;

(5)

~ iT ±2dt .

Here is a typical result. Proposition 1. Consider the autonomous system:

x + av(x) :3 0

(6) where V : IRn

(7)

~

IR is convex and satisfies: V(x) > V(O) = 0

'Ix =I- 0

134

(8)

III. Fixed-Period Problems: The Sublinear Case

V(x) 2: v (1Ixll)

(9)

3k

with V(S)S-l

> 0, 3c

as s

-+ 00

k

2

211xll + c

V(x) 2:

-+ 00

' k

K := lim inf V(x) x-+O

.

Then, for all T such that 27rK- 1 j2

(11)

< T < 27rk- 1j2

problem (7) has a non-trivial T -periodic solution. B. BoIza Problem We consider the boundary-value problem of Example H.4.1:

q = H~(q,p) { P = -H~(q,p)

(12)

q(O)

=

qo,

q(T)

= ql .

After a suitable change of variables, this problem can be rewritten as:

q = H~ (q + +(ql - qo),p) { p = -H~ (q + +(ql - qo),p) q(O) = 0,

q(T) = 0 .

We gave the direct functional as:

and the dual as: (14)

tJt(p, q) :=

iT

[-pq +

pq + H* (q + ~, -p)] dt .

It should be noted that

(15)

tJt(p + ry, q) = tJt(p, q)

so that we can replace p by 15:

(16)

15(t) := p(t) -

thereby enjoying Wirtinger's inequality

' for some 0: > 2. The dual action 'ljJ can no longer be minimized (it is easy to check that inf'ljJ = -00), so a subtler procedure is required. It will be described in the opening section.

1. Mountain-Pass Points The aim of this section is to find critical points of functionals in some simple situations where minimization procedures are not available. The proof will make repeated use of the following variational principle of Ekeland: Theorem 1. Let (X, d) be a complete metric space and iP : X ---+ 1R U {+oo} a lower semi-continuous function which is bounded from below: inf iP > -00 .

(1)

Let z E X and

f

2: 0 be such that: iP(z) ::; inf iP + f

(2)

Then there is some point y

(3) (4)

iP(y) Vx EX,

E

.

X such that:

+ fd(y, z) ::; iP( z) iP(x)::; iP(y) - fd(x, y) .

As an immediate consequence of inequality (3), we have:

(5)

d(y, z) ::; 1

o

1. Mountain-Pass Points

tJ!(y) :s; inf tJ! + €

(6)

137

•

Note also that we can define a new distance on X by taking 8(Xl' X2) = A-ld(Xb X2) for some fixed A > O. Then (X,8) is still a complete metric space, with the same property as (X, d). Applying Theorem 1 to (X,8) gives us a point Y>. E X such that:

(7) (8)

'\Ix EX,

For instance, if this case:

€

> 0, we may take A = .;E. Let us rephrase the result in

Corollary 2. Let (X, d) be a complete metric space, and tJ! : X -+ IR U {+oo} a l.s.c. function on X with inf F > -00. Let € > 0 be given, and choose z E X such that:

tJ!(z) :s; inf 0 such that 8(Xl,X2) ~ 'Y Ilxl - x211 whenever Xl and X2 belong to B. By radial symmetry, we find that, when Xl = 0, the infimum on the righthand side of formula (22) must be achieved by the line segment from 0 to X2 = x. Hence:

(23)

8(0, x) =

1 +IlxllIlxll 1

0

1

t

dt = Log (1

+ IlxlD .

It follows that 8-bounded and norm-bounded subsets of X are the same. Any Cauchy sequence for 8 is 8-bounded, and hence norm-bounded. By the preceding considerations it is also a Cauchy sequence for the norm, and hence norm- and 8-convergent. So (X, 8) is a complete metric space. Applying Theorem 1, we get:

1. Mountain-Pass Points

139

Corollary 4. Assume X is a Banach space and P : X --+ lR is Gfiteauxdifferentiable, l.s.c. and bounded from below. Then there exists a sequence Xn such that:

(24) (25) Proof. Apply Corollary 2 to (X, 8) with that

€

=

;2' We get a sequence

o Xn

such

(26) (27)

' 0 and U

EX:

(28)

8: (29)

Devide both sides by t, and recall the definition of the geodesic distance 1

1

lit

- (p(Xn + tu) - p(Xn)) ~ --Ilullt n t Letting t

--+

0

(1 +

ds

II Xn + su II)

0, we get:

(30) Changing u to -u, we get the reverse inequality, as in Corollary 3, and hence:

(31)

o

from which the result follows immediately.

Of course, the existence of sequences Xn satisfying (14) or (25) relies heavily on the fact that P is bounded from below. We now turn to another situation where such sequence exist: Definition 5. A closed subset F separates two points Zo and Zl belong to disjoint connected components in X \ F.

Zl

in X if

Zo

and 0

Since X \ F is locally connected, any component will be an open and closed subset of X \ F, and hence open in X. Denoting by no the component containing zo, and by n1 the union of all components not containing zo, we get a partition of X \ F into two disjoint open sets, with Zi E ni , i = 0,1.

140

IV. Fixed-Period Problems: The Superlinear Case

Theorem 6 (Ghoussoub-Preiss). Let X be a Banach space and ip : X ---t IR a continuous, Gateaux-differentiable function, such that ip' : X ---t X is continuous from the norm topology of X to the weak* topology of X*. Take two points (zo, Zl) in X and consider the set r of all continuous paths from Zo to Zl:

(32)

r:= {c E Co ([0, 1]; X) Ic(O) = Zo, c(l) = Zl} Define a number '"Y by:

(33)

'"Y := inf max ip (c(t)) cEro::;t::;1

Assume there is a closed subset F of X such that:

(34)

F

with ip"(:= {x E X

n ip"(

I ip(x)

separates Zo and

Zl

2: '"Y}. Then there is a sequence

Xn

in X such that:

(35) (36)

(37)

D

Proof. Write F"( := ip"( n F. Since F"( is a closed subset separating Zo and Zl, we may partition X \ F"( into two open subsets flo and fl l , with Zi E fli for i = 0,1. Choose E such that:

(38) and find a path c E

(39)

r

such that: E2

Max ip (c(t)) < '"Y + -4 .

O::;t9

Such a path c always exists because of the definition of '"Y. We then define two numbers to and tl, with 0 :::; to < tl < 1, by:

(40)

to = sup{t E [O,l]lc(t) E flo, 8 (c(t),F"() 2: E}

(41)

tl = inf(t E [to, l]lc(t) E fll,8(c(t),F"() 2: E}

Note that 8 (c(t),F"() :::; E for to:::; t:::; tl. Consider the space:

and endow it with the uniform distance

1. Mountain-Pass Points

141

d(h, h) = Max 8 (fl(t), h(t))

(43)

to::;t::;tl

Set: (44)

and define a function cp : r(to, tt}

-+

1R by:

cp(f):= Max (p(f(t))

(45)

to::;t::;tl

+ tJt(f(t)))

Since J(t o) = c(t o) E no and J(tl) = c(tt) E nl, there must be some point tf E (to, it) where J(tf) E ana c F y, and hence: (46) for every J E r (to, tt). On the other hand, consider the restriction path c to [to, ttl. We have:

(47)

c of the

cp(c):::; Max {p(c(t))+tJt(x(t))}:::; ('Y+ £2) +£2. 4

o::;t::;1

r(to, tt} is a complete metric space, and the function cp is l.s.c. and bounded from below. We have found a point c where cp(C) :::; Infcp + £2/4. We may therefore apply Corollary 2 to find a path E r(to, tt} such that:

1

Now introduce the set M consisting of all points t E [to, tIl where (p+tJt)ol attains its maximum:

(50)

M

:=

{t

E

[to, tilip (f(t))

+ tJt (f(t))

=

cp(f)} .

It is compact, non-empty, and it contains neither to nor it. Indeed, from the definition of to and it it follows that 8 (C(ti), = £ for i = 1,2, so that 7jJ (C(ti)) = 0 and hence:

FI)

(51)

p (j(ti)) + llt (j(ti)) :::; p (C(ti)) + tJt (C(ti)) :::; 'Y + :

(1)

which is strictly less than cp by inequality (46). I claim that M contains a point t where:

(52)

IV. Fixed-Period Problems: The Superlinear Case

142

Assume otherwise, that is:

(53) It follows that for every t E M there will be a vector u(t) E X such that

(54)

IIu(t)II = (l+llf(t)I!)-l

(55)

(pI (f(t)) ,u(t)) < -

~€

•

By the continuity assumption on pI, there will be an open neighbourhood 8 E N(t), we have:

N(t) of t in [to, t11 such that, for every

~ ) (pI ( /(8) ,u(t))
0, and we may substitute hv for / in formula (49). We get:

1+

(62)

Vh

> 0,

Max {p(zo),p(zt}} .

(85)

Then there is a critical point of P on the level 'Y

(86)

p'(x) = 0 and p(x) = 'Y .

:3x:

o

Proof. Take F = X in Theorem 6. Assumption (85) implies that P,¥ separates and Zl, and the result follows. 0

Zo

Then a situation when equality may hold in (85). Defining X, P, rand 'Y as in Corollary 9, we have

Corollary 10. Let F c X be a closed subset separating

Zo

and

Zl.

Assume that:

(87) I::/xEF,

(88) (89)

p(x)2::'Y

P satisfies condition (C) on the level 'Y for the set F .

Then F contains a critical point of P on the level 'Y:

(90)

:3 x E F:

P' (x),

p(x) = 'Y .

o

The existence of a critical point x will not be enough for our purposes. The feeling we get from our construction is that x should be some type of saddle-point, or mountain-pass. It is this kind of information we are after, and it turns out that Theorem 6 provides it. First some notations and definitions. We define X, P, r, 'Y, F as in Theorem 6, and we assume that P satisfies condition (C) for F on the level 'Y. Set:

(91)

K: = {x E K I p(x) = 'Y,p'(x) = O}

(92)

M: = {x

E

K I x is a local minimum for p}

146

IV. Fixed-Period Problems: The Superlinear Case

It follows from condition (C) that K

nF

is compact and non-empty.

Definition 11 (Hofer). Let x E K, so that 'Y = 4>(x). We say that x is a mountain-pass point if, for every open neighbourhood U of x in X, the set

(93)

4>'it:= {x

E

UI 4>(x) < 'Y} o

is neither empty nor connected.

Saying that 4>'it is never empty means that x is not a local minimum. Note also that since 4>'it is open in X, it is connected if and only if it is path connected. For U = X, we shall write 4>"( instead of 4>'it:

4>"( := {x I 4>( x) < 'Y} .

(94)

For the sake of brevity, we shall henceforth write mp instead of mountainpass. Introduce another subset of K:

(95)

P := {x E K I x is a mp-point for 4>}

The following theorem tells us that K contains either a mp-point or a local minimum. Theorem 12. Assume that Fn4>"( separates (CJ at the level 'Y. Then:

(96)

F

n M -# 0

or

F

Zo

and Zl and 4> satisfies condition

n P -# 0 .

Proof. As before, noting that F"( := F n 4>"( separates Zo and Zl, we find two open sets no and n1 with no u n1 = X \ F, no n n1 = 0, and Zi E ni . Suppose F n P = 0, that is, F contains no mp-point. We claim that there is an 1:1 > 0 such that the set

(97) meets only finitely many components of 4>"( say U 1 , ... ,UN. Indeed, otherwise we could find a sequence Xn in F"( n K and a sequence Un of pairwise disjoint components in 4>"( such that 8 (xn' Un) --t O. But then, any cluster point x of the sequence Xn would be a mp-point belonging to F"( c F, which contradicts our initial assumption. Consider now the sets Qi, 1 SiS N, defined by:

(98) Again, any point in Qi n (U#iUj there is an 1:2 S 1:1 such that:

(99)

)

would be a mp-point. It follows that

1. Mountain-Pass Points

E E

We may of course assume that (0, E2) we associate the set

(100)

R(E):=

E2

147

< 8(Zi' Fy) for i = 0,1. With every

[noU U{x 18(x,Qi) < E}l U{x 18(x,Qj):::; E} , iEA

jEB

where the set of indices i = 1, ... ,N is partitioned into:

{i I

U; c no}

(101)

A:

(102)

B:={ilu;nno =0}

=

R(E) is an open subset of X containing zo, while Zl (j. R(E). Its boundary 8R(E) therefore separates Zo and Zl. It follows from Definition (100) that:

uU; uU;

(103)

C R(E)

iEA

(104)

C

X\R(E)

iEB

and hence: (105)

8R(E)nq>'Y =8R(E)n

(QU;) =0.

Consequently, P ~ 'Y on R( E). Therefore, we may use Theorem 6 to find a critical point x€ for P in 8R(E): x€ E Kn8R(E).

(106)

We claim that, for each E, the point x€ is a local minimum for P. Indeed, if such is not the case, then x€ E U; for some i E {I, ... , N}. If i E A, then x€ E Qi C R(E), and since R(E) is open, Xe (j. 8R(E), yielding a contradiction. If i E B, then Xe E Qi eX \ R(E), yielding Xe (j. 8R(E), still a contradiction. Finally observe that 8 (xe, F"() -7 0 when E -7 0 and since K is compact, this shows that F"( n Me =I 0, as announced in formula (95). 0 Taking F = X, we get an earlier result of Hofer: Corollary 13. Let X, P,

(107)

K

r, 'Y

be as in Corollary 9. Then:

n (M \ M) =I 0

or

K

n P =I 0 .

o

In other words, on the level 'Y there is either a mp-point, or a point which is not a local minimum, but which is the limit of a sequence of local minima on the level 'Y.

148

IV. Fixed-Period Problems: The Superlinear Case

Proof. Consider the set:

(108) Let F be its boundary, and apply Theorem 12 to F. We get F n M =I- 0 or F n P =I- 0. On the other hand, any local minimum of P on the level 'Y must be interior to P'Y' so F n M = 0. The result follows. D Notes and Comments. The chronology is as follows. Palais and Smale introduced their celebrated condition (PS) to extend Morse and LinsternikSchnirelman theory to the infinite-dimensional setting (see [PalS], [Pall]' [PaI2]). Using the deformation lemma (see Sect. V.2 for an equivariant version) they were then able to give results on the existence of critical points, provided the functional was regular (slightly better than C 1 ). That condition (C) is enough was first remarked by Cerami in [Cer]; see [Ben2] for other weights. Then came the celebrated theorem of Ambrosetti and Rabinowitz [AmbR] (here Corollary 9), which gave a simple procedure to find a critical point which is not a local minimum. They had, of course, more stringent regularity conditions than we assume here. Theorem 1 is due to Ekeland ([Eke1], [Eke2]) and throw a new light on the whole thing. One could avoid the deformation lemma, and weaken considerably the regularity assumptions. Theorem 12 is due to Hofer ([Hof2] and [EkeH1]), relying on the deformation lemma. Ghoussoub and Preiss, in [GhoP], found a way to localize these results of Ekeland's variational principle. Theorem 6 is due to them, and has among other consequences several earlier results of Pucci and Serrin ([PucS1]' [PucS2]) On the topological structure of the set of critical points.

2. A Preliminary Existence Result This section serves as an introduction to superquadratic Hamiltonians, and to set up the moutain-pass situation which will be exploited further in the following sections. We will prove an existence result for periodic solutions which will be considerably improved later on. Theorem 1. Let H : [0, T]

(1)

(2)

Vt

E

[0, T],

X

lR2n

->

lR be such that:

H(t,.) is strictly convex and C 1 on lR2n

H and H' are continuous on [0, T]

Assume that:

(3)

V(t, x) ,

H(t,x)

and that there are constants a > 2 and W

(4)

3r > 0:

H(t,AX)

2:

AQ:H(t,x)

2: H(t,O) E

=

X

lR 2n

.

°

(0, 27rT- 1 ) satisfying:

for

Ilxll2: r

and A 2: 1

2. A Preliminary Existence Result

(5)

lim sup H(t,x) o::;t::;T,lIxll---+0 :

H(t,x) :::;

W

"2llxll

149

Ilxll- a < 00 2

Ilxll:::; 10

for

•

Then, the boundary-value problem {

(7)

X = JH'(t, x)

X(O)

= x(T)

o

has a non-trivial solution.

By assumption (3), the trivial solution x = 0 (equilibrium at the origin) is always present. The theorem asserts the existence of another one provided Tw < 27r. Condition (6) gives us the behaviour of H near the origin. For instance, if (8)

H(t,x)

IIxll- 2 -70

Ilxll--- 0

when

then it is satisfied for every w > 0; and in particular for w < 27rT- 1 . In the autonomous case, this gives us an existence result for every T > 0: Corollary 2. Let H E C 1 (lR2n, lR) be a strictly convex function such that:

(9) (10)

Vx,

3r> 0

H(Ax)

(11)

2 H(O) = 0

H(x)

2 Aa H(x)

for

lim sup H(x) Ilxll---+ 2, and

(12)

lim H(x) IIxll---+O Then, for every T

Ilxll- 2 = 0

.

> 0, the system

x = JH'(x)

(13)

o

has a non-trivial T -periodic solution.

Of course, conditions (3) and (6) have dual versions in terms of the Fenchel conjugate H*(t; x) of H with respect to x, for fixed t as in formula (11.3.10). Lemma 3. We have:

(14)

V(t,y),

H*(t,y) 2 H*(t,O)

=0

150

IV. Fixed-Period Problems: The Superlinear Case

(15)

317 > 0 :

H*(t, y)

2:

1

2w

lIyl12

lIyll:::; 17 .

for

Proof. Straightforward. We have:

(16)

17:= Min

{IIH'(t,x)llllIxll 2: E}

D

Condition (4) tells us that H(t,x) grows faster than Setting

ma := Min {H(t,x) a

(17)

-

10:::; t:::; T, Ilxll =

Ilxll a

at infinity.

r} > 0

we have:

(18)

H(t, x)

2:

ma a

IIxr r

Ilxll 2: r

for

.

It is equivalent to a condition on the derivative, which will be extremely useful in the sequel: Lemma 4. Condition (4) is equivalent to the following:

(19)

(x,H'(t, x))

2: aH(t,x)

for

Ilxll2: r.

Proof. Assume (4) holds. Fix (t, x) with Ilxll 2: r and consider the functions CXl

Ilyll-f3 > 0

from wich similar estimates will follow: Mf3 (3Rf3

(32)

(33)

f3

IIYII

*.

1

~ H (t, y) ~ (3k f3

IIVW(t,y)1I ,;

lIyll

f3

for

Ilyll

~ [G~)" -:, ]"ylI'-- ,

where R has been defined in Lemma 5. The existence of k taken arbitrarily large) follows from condition (11), and M := Max {H*(t; y)lllyll = Rand

(34)

~ R

> 0 (which can be

0:$ t :$ T}

.

Taking condition (15) into account, we find that we can choose k so large that: {

(35)

H*(t; y)

~ 2~ IIyl12

H*(t; y) ~ f3!i3

Ilyllf3

for for

IIYII:$ TJ lIyll ~ TJ

•

We now introduce the dual action functional. The Hamiltonian H clearly satisfies conditions (51) to (54) of Sect. 11.4. By Proposition II.4.6, problem (7) then is equivalent to finding critical points of the functional

(36)

!Ii(x):= iT

G

(Jx,x)

+ H*(t, -Jx)] dt

on the space: (37)

W 1 ,f3 := {x E Co ([0, T]; IR2n) I x E Lf3, x(O) = x(T)}

As in the preceding chapter, we take advantage of the fact that !Ii is invariant by space translations:

(38) to perform the change of variables (39)

1j;(u)

:=

iT

x=

u. We get a reduced functional:

[~(JU,JIu) + H*(t, -Ju)] dt

2. A Preliminary Existence Result

153

on the space:

(40) Here IIu is defined to be the primitive of u with mean value 0:

!

(41) Lemma 7. II : L~

-t

and

(IIu) = u

iT

(IIu)dt = 0 .

L~ is a compact operator.

Proof. This was proved in Proposition III.1.2 when case, write II u explicitly: (42)

(IIu)(t) =

it 0

u(s)ds -

O!

liT it

T

0

dt

0

= 2 = (3. For the general

u(s)ds.

Now let Un converge to u in L{3. Then IIUn (t) is clearly uniformly bounded for 0 :::; t:::; T and n E IN. By Cauchy-Schwarz, we have

so that IIu n , n E IN, is an equicontinuous family of functions. By Ascoli's theorem, it must be relatively compact in Co, and hence in £ cp(O, 0) = 0 .

2. A Preliminary Existence Result

155

Finally, we have: (55) where the left inequality comes from IlvlL:x> :::; '" and the right one from Holder's inequality. That is, the L 2 -norm and the Li3- norm for v are equivalent. It follows that, if Ilull i3 = p is small enough, the pair to ./II, and formula (48) follows with (56)

0 < a := Min { A(t, x) > 0 .

Then every non-trivial T-periodic solution of Eq. (1) is admissible in the sense of Definition 1.6.1. We recall that the index iT(X) is defined to be the dimension of the negative subspace associated with the quadratic form qT on L~ given by

(4)

qT(U, u) : = iT ~ [(Ju, JIu) = iT ~ [(Ju,JIu) o

2

+ (H"(t,x(t))-lJu, Ju)] dt + (I(H*)I" (t, -Ju(t))Ju, Ju)] dt

and that the nullity VT(X) is the dimension of the kernel of qT (see Sect. I1.6). The problem is now to relate qT and '¢. Unfortunately, qT is not the Hessian of'¢ at u; they are not defined on the same spaces (L~ and L~ respectively, with 1 < (3 < 2), and anyway '¢ is not C 2 • The regularity of H* is not in question; it is simply the fact that, since (3 > 2, if we differentiate twice we fall outside the range of Krasnoselskii's Theorem II.3.4. Put in another way, the integral (4) does not define a continuous quadratic form on Lj3, for (3 < 2. On the other hand, the restriction of'¢ to L': c L~, denoted by '¢=, clearly is C 2 • If u is a critical point of '¢ on L~, then u = : and x solves Eq. (1), so u E Co c L =. The Hessian of '¢= at u is the restriction of qT to L o=·.

(5)

('¢::a(u)v, v) = qT(V, v) 'ltv E L':' . So we wish to bring the whole mountain-pass situation down from L~ to

L~. It is the purpose of the following lemma.

Lemma 1. Take u E K(,¢, ')'), that is, '¢'(u)

(6) (7) (8)

= 0 and '¢(u) = ,)" and set

'¢'Y:={UEL~I,¢(u) 0, there exists a 8 > 0 such that, whenever'P is a path component of'ljJ'Y with 'P n B {3 (u, 8) =J. 0, then:

(9)

D

In other words, if a path component of'ljJ'Y comes arbitrarily close to U in L~, then it comes arbitrarily close to U in L':.

Proof. Let 'P be a path component of'ljJ'Y with 'P n B{3(u, 8) =J. 0, and set:

(10) Since 'P

C

'ljJ'Y, it follows immediately that:

(11)

"/6,P

< "/ .

The proof now proceeds in several steps.

Step 1: The infimum is achieved at a point U6,P Take a minimizing sequence Un E 'P n B{3(u, 8). We have 'ljJ(un ) ---* ,,/o,p. Since L~ is reflexive, we may choose the sequence to be weakly convergent:

(12)

un

---*

u

for a (L~, V")

.

We wish to show that U E 'PnB{3(u, 8). Since the ball is convex and closed, it is clear that U E B{3(u,8). Since 'ljJ is weakly l.s.c. (remember that it is the sum of a compact term and a convex term), we have

(13)

'ljJ(U)

~

lim inf 'ljJ(un) =

"/6,P .

We claim that there is some no such that the line segment connecting u to

uno:

(14)

{hu + (1 - h)unoiO

~

h

~

I}

lies entirely inside 'ljJ'Y. This will prove that u lies in the same path component as Uno, that is u E 'P. Argue by contradiction, assuming that there are sequence hk in [0,1) and Un(k) in 'P n B(3(u, 8) such that:

(15) We write this explicity:

(16)

160

IV. Fixed-Period Problems: The Superlinear Case

Remember that H*(t,·) is convex, so that:

(17)

Because of (12), we have Vk ----+ U for (j (L~,LO:), and hence IIVk ----+ IIu strongly, since II is compact. We may assume that hk ----+ hoo in [0,1], and we know that 'I/J (Un(k)) ----+ 'Y6,P. Passing to the limit in inequality (17), we get: (18)

that

Using (13), we get 'Y :S 'Y6,P, which contradicts (11). This finally proves E 'P and concludes Step 1.

u

Step 2: The necessary conditions for optimality at U6/P

'P is an open subset of L~, so u is in fact a local minimum of 'I/J on the ball Bf3(u, 8). Since'I/J is C 1 , and the ball is defined by the constraint Ilu - ull f3 :S 8,

the left-hand side of which is C 1 , there is a Lagrange multiplier A = A6,P E lR such that:

(A (u - u)f3- 1 + 'I/J'(u) , w) = 0 Vw E L~ .

(19)

Here vf3 -1 denotes the function v II v II f3 - 2 . The Lagrane multiplier A cannot be negative. Indeed, since 'P is open and Bf3(u,8) is convex, the point (1 - h)u + hu must belong to 'P n Bf3(u, 8) for small h. Hence:

d~ 'I/J ((1- h)u + hu) =

(20) Taking w = (21)

u-

('I/J'(u) , u, -u) ::::: 0 .

u in Eq. (19), we get

A IIu

-

ull f3

+ ('I/J'(u),u -

u) =

o.

Comparing with (20), we derive: (22) Now replace 'I/J'(u) by its value (Lemma 2.8) in formula (19). It becomes:

(23)

(A(U-U)-l-JIIu+J\lH*(t,-Ju),w) =0 VWEL~.

3. The Index at Mountain-Pass Points

This means that there is a constant vector A (u -

(24)

u)/3- 1 -

e=

fo,p

E ]R2n such that:

f

JIIu + J\l H* (t, -Ju) =

161

a.e.

On the other hand, we know that 'ljJ' (u) = 0, so that there exists ~ E ]R2n with:

(25)

-JIIu+J\lH*(t,-Ju) =~

a.e.

Substracting both equations, we get:

(26) A (u -

u)/3- 1 + J\lH* (t, -Ju) -

J\l H* (t, -Ju) - JII (u - u) =

Step 3: An L oo estimate for uo,p when 8

~

f-~.

0

We first derive a uniform estimate for [o,p. Integrate both sides of Eq. (26) against u - u, and then divide by !lu - ull/3. We get (not writing the variable

t):

A Ilu -

(27)

ullg- 1 + (J\l H*( -Ju) -

J\l H*( -Ju), u - u)

- (JII(u - u), u - u)

Ilu -

Ilu -

ull~l

=0.

ull~l

By Krasnoselskii's theorem 11.3.4, since \l H* satisfies the estimate (2.33), the map u ~ \l H* ( - J u) is continuous from L/3 to LOI.. The second term on the left therefore goes to zero when 8 ~ O. The third term on the left also vanishes when 8 ~ o. So the remaining term has to vanish too. More precisely, if 8n ~ 0 and 'Pn , n E IN, is a sequence of path components in 'ljJ'Y such that B/3 (u, 8n ) n 'Pn -I- 0, then:

(28) Writing this back into Eq. (26), and using Krasnoselskii's theorem again, we find that

(29) This will give us an UX) estimate for UOn,Pn. Indeed, simplify the notations:

(30)

Un := UOn,P n ,

-

-

en:= eOn,Pn,

An:= AOn,Pn

and rewrite Eq. (26) as follows: J\lH* (t, -Jun ) + An (Un -

(31) (32)

fn(t) := J\lH* (t, -Ju(t))

u)/3- 1 =

+ JII (Un -

u)

fn

+ fn - ~ .

The fn are continuous and converge uniformly to J\lH* (t, -Ju(t)) as n~oo.

Note that the left-hand side of Eq. (31) is the derivative at y = u(t) of the convex function:

162

IV. Fixed-Period Problems: The SuperIinear Case

y - t H* (t, -Jy) +

(33)

>; lIy -

U{t) III'

.

Hence, writing that the function lies above its tangent hyperplane at u(t):

H*{t; -Ju{t)) 2: H* (t; -Jun{t)) +

(34)

+ (fn(t), u{t) -

>;

lIun(t) - u(t) III'

Un (t)) .

Using estimate (2.32), we get for every t E [0, T] and n E 1N the alternative

(35)

Ilun{t)ll:::; R

either or H*(t, -Ju(t)) 2:

(36)

f3~1'

Ilun{t) III' + (fn{t), u{t) - Un{t))

from which it follows that the Un are essentially bounded:

(37) Step

3A:

4: Uo,lP

lIun{t) I

converges to

:::; A

u in Loo

Vt E [0, T]

Vn E 1N .

when D - t 0

We claim that:

(38)

[Iun -

ull oo - t 0

when n

- t 00 .

Assume otherwise. Then we can find a subsequence Un', and a sequence

tn'

E [0,1]' such that

(39)

Un' (t n,) - u{tn')

-t

Y "# 0

when n'

- t 00 .

Since An 2: 0, inequality (34) will imply in the limit:

(40)

H* (t; -Ju{t)) 2: H* (t; -Jy) + {J'VH* (t; -Ju(t)) ,u(t) - y)

which we rewrite as:

(41)

H* (t; -Ju(t)) - H*{t; -Jy) 2: ('VH* (t; -Ju(t)) , -Ju{t) + Jy) On the other hand, since H* is strictly convex with respect to y, we have

(42)

H* (t; -Jy) - H* (t; -Ju{t)) > {'V H* (t; -Ju{t)) , -Jy + Ju{t))

whenever

y "# O. Adding up the two last inequalities, we get a contradiction.

Step 5: Conclusion

Assume Lemma 1 is false. Then there is an E > 0, a sequence Dn a sequence P n of path components of'ljJr such that:

-t

0 and

3. The Index at Mountain-Pass Points

(43) (44)

163

n B(3("u, c5n) =I- 0 'Pn n B(3(u, c5n ) n Boo(u, 10) = 0 . 'Pn

Because of condition (43), we find in 'Pn n B(3(u, c5n ) a minimizer Un for 'l/J. By the preceding analysis Ilun - ulloo - t 0 when n - t 00, which contradicts condition (44). 0 Lemma 1 has several interesting consequences. Corollary 2. u E Lr:: is a local minimum for 'l/Joo if and only if it is a local minimum of 'l/J.

Proof. If u E Lr:: is a local minimum for 'l/J, it clearly is a local minimum for 'l/Joo. Conversely, assume u is not a local minimum for 'l/J. Then'l/J"Y has points in L~ arbitrarily close to u. It then follows from Lemma 1 that it contains points in Lr:: arbitrarily close to U, which means that u is not a local minimum for 'l/Joo. 0 Corollary 3. u E Lr:: is an isolated critical point for 'l/J if and only if it is an isolated critical point for 'l/Joo.

Proof. If Un E Lr:: is a sequence of critical points of 'l/Joo converging to u in Lr::, they also converge in L~. Conversely, if Un E L~ is a sequence of critical points of'l/J converging to u in L~, they are all in Lr::, and a simple adaptation of the proof of Lemma 1 (where the Lagrange multiplier A disappears) shows that in fact they converge in L r:: . 0 Corollary 4. Ifu E Lr:: is a mp-point for 'l/J, then it is a mp-point for 'l/Joo.

Proof. Assume u is not a mp-point for 'l/Joo. Then, either it is a local minimum for 'l/Joo, or there is a neighbourhood V of u in Lr:: such that V n 'l/JJo is pathconnected. In the first case, u will also be a local minimum for 'l/J, and so cannot be a mp-point for'l/J. So assume we are in the second case. Choose 10 > 0 so small that Boo (u,€) C V, and apply Lemma 1 to get a corresponding 15 > O. Let 'PI and 'P2 be two path components of'l/J"Y such that 'Pi n B(3(U, 15) =I- 0, i = 1,2. Then we can find for i = 1,2:

(45) Since Vn'l/JJo is path-connected in Lr::, it is also path-connected in L~. So and U2 belong to the same path component of 'l/J"Y. It follows that 'PI = 'P2 , that is, there is only one path component 'P in 'l/J"Y intersecting B(3(u, 15). Set: UI

(46)

W:= B(3(u, 15) U'P .

Then W is a neighbourhood of u in L~, and W n 'l/J"Y connected, so u cannot be a mp-point for u.

'P is path-

o

164

IV. Fixed-Period Problems: The Superlinear Case

We are brought back to the study of the C 2 function 'l/Joo. We have 'l/J'oo(u) = ~ E lR2n. By Proposition 1.4.2, the Hessian 'l/J'/x,(u) = qT induces an orthogonal splitting of L~ (47)

Since E_ and Eo are finite-dimensional, this induces a topological splitting of L':: (48) where E'f := E+ n L':. We denote by P _, Po and P+ the corresponding projections, and by u_, U o and u+ the components of u on E_, Eo and E+. Lemma 5. There is a neighbourhood V of u_ 0" :

V - t E'f such that

+ Uo in E_ EEl Eo

('l/J'oo(v + O"(v)), w) = 0 'rIw

(49)

E

and a C1 map

E'f

(50)

The composed map ¢oo(v) := 'l/Joo(v + O"(v)) is C 2 • Proof. Define a map S : L':

-t

L': by:

S(u) := -JIIu + J'VH*(t, -Ju) -

(51)

D

~ iT J'VH*(t, -Ju)dt

so that: (52)

('l/J'oo(u), v) = iT (S(u), v) dt . Clearly S is a C 1 map. We have

(53)

(Sf(U)V,W) = qT(V,W)

'rIv E L~ 'rIw E L~

which implies that Sf (u) preserves the splitting (48). In particular, we have

(54)

Sf (u)E'f

C

E'f .

Sf (u) induces an isomorphism of E+ onto itself. Using some regularity, we can show that Sf (u) is a bijection of E'f onto itself, and it then follows from the open mapping theorem of Banach that Sf (u) is an isomorphism of E'f into itself. Now define a map S+(u_ + uo,·) of E'f into itself, depending on the parameters u_ E E_ and U o E Eo, by: (55)

3. The Index at Mountain-Pass Points

165

It is CI, and we just saw that 8~ ('it-, u o, u+) is an isomorphism. On the other hand, since u is a critical point of 'l/Joo, we have 8'(u) = 0, which implies:

(56) It then follows from the inverse function theorem that there is a neighbourhood V of u_ + Uo in E_ E9 Eo and a unique C 1 map a : V --t Ef such that

(57)

8+ (v,a(v)) = 0 Vv E V

(58) The first equation is also 8 (v + a(v)) E E'f, which is equivalent to (49) because of relation (52). Differentiating it at u_ + uo , we get:

(59) We already noticed that E_, Eo and E'f are invariant subspaces of 8'(u). This means that 8' (u)v has no component in Ef, and the preceding equation amounts to:

(60) Since 8' (u) induces an isomorphism of E'f onto itself, this boils down to a' (u_ + uo ) = 0, as announced. It remains to show that ;[00 is C 2 . It is C 1 since a is C 1 • Compute its derivative at v E E_ E9 Eo: (;['oc,(v),w) =

(61) =

iT iT

(8(v

+ a(v)),w + a'(w))dt

(8(v

+ a(v)), w) dt

since 8(v + a(v)) E E_ E9 Eo and a'(w) E E+. Hence

(62) which is again C 1 .

D

We now need to put ;[00 in standard (or normal) form near the critical point u. This is the purpose of the next result, which includes the classical Morse lemma and the Gromoll-Meyer theorem as special cases: Normal Form Theorem. Let U 30 be an open subset in a Hilbert space V and cp : U --t lR a C 2 function with cp'(O) = o. Assume that L := cp"(O) is Fredholm, so that V splits orthogonally into positive, negative, and null subspaces relative to cp" (0):

166

IV. Fixed-Period Problems: The Superlinear Case

with Eo = Ker Land E+ EB E_ = L(V). Then there exists an open neighbourhood V of 0 in L(V), and open neighbourhood W of a in Ker L, a homeomorphism h from V x W onto an open neighbourhood of a in V, with h(O, 0) = 0, and a C 2 function f : W - t IR such that: (64)

the restrictions of h(·,O) to V is a CI-diffeomorphism

(65) (66)

1'(0)

O.

'IjJ'Jo for t > 0, and

E (0,1] .

In addition, we know that

c'(O)

(89)

E

E_ \ {O} .

Because of formula (47) and the definition of E_, this implies that:

(90)

('IjJ~(u)c'(O),

Now consider the path

(91)

c'(O)) < 0 .

c defined by: c(t)

:=

u + tc'(O) .

We claim that there is some p > 0 such that, when 0 < t < p, the line segment starting at c(t) and ending at c(t) lies entirely within 'IjJ'Jo. Suppose otherwise. Then there are sequences tn ----* 0, and An in (0,1] such that for every n: (92)

that is

3. The Index at Mountain-Pass Points

Remember that

169

w'oo (u) = 0, so that this inequality implies in the limit: (w~(u)c'(O), c'(O)) ~

(93)

0

which contradicts inequality (90). We have thus proved that there is some p > 0 such that

o < t < p =;. c(t) "-' c(t)

(94)

Remembering relation (88), we see that c(t) concludes the proof.

. rv

U

for 0 < t < p. This 0

Corollary 8. If dimE_ ~ 2, then there is a neighbourhood U ofu in L': such that wJo n L': is path-connected.

Proof. Take the neighbourhood U defined in Lemma 7. Let points in Un wJo. We have seen that Ui rv u~, i = 1,2, with

Ul

and

U2

be two

(95) (96) for some w~ E E_ \ {O}. If dimE_ ~ 2, the space E_ \ {O} is path-connected, so that we can find a continuous path w=-, 1 :::; t :::; 2, connecting w~ and w~, with

We define accordingly: U~ := u + h (w=-)

(98)

+ (it + h(w=-)) (7

and the normal form yields immediately:

(99) whereby we have shown that

(100) Hence

Ul "-' U2,

o

and the corollary is proved.

We now sum up the results of this section. Theorem 9. Let u E K (W, 'Y) be a mp-point, and let T -periodic solution of problem (1). Then

(101)

iT(U) :::; 1 .

x

be the corresponding

170

IV. Fixed-Period Problems: The Superlinear Case

If iT (x) = 1, then '¢'Y has exactly two path components 'Po and 'Pi. There is an eigenvector e =I=- 0 associated with the negative eigenvalue of '¢" (u), and we can choose t > 0 so small that:

(102)

u + te E 'Po

(103)

u-

te E 'PI .

Proof. Suppose u is a mp-point for '¢. Then it is a mp-point for '¢OO by Corollary 4, and we must have dim E_ < 2 by Corollary 8. But

(104) by definition. Hence formula (101). Define I: and 8 as in Lemma 1. Since u is a mp-point for ,¢, there are two distinct path components 'Po and 'P1 of '¢'Y intersecting Bf3(u, 8), and they also intersect Boo (u, 1:). Take: (105)

Ui

E

'Pi

n

Bf3 (u, 8)

n

Boo (u,l:) ,

i = 0, 1 .

Now let U be the neighbourhood of u in Lr:' defined in Lemma 7. Since we are free to choose I: > 0, we may assume that Boo (u,l:) C U. By Lemma 7, we know that for t > 0 small enough, (106)

Uo

rv

U + te

(107)

U1

rv

U - te

where the relation rv means "belong to the same path-component of '¢Jo in Lr:''', and hence of '¢'Y in L~. If u + te and u - te belonged to the same path-component 'P of in L~, then so would U1 and U2 by the relation (106), (107), and we would have (108) contradicting the fact that 'Po and 'P1 are distinct. If there is another path-component 'P2 intersecting Bf3(u,8), we get a point (109) which, by Lemma 7, turns out to be equivalent to Uo or U1. SO 'P2 = 'Po or

'P2

=

'Pl.

0

Notes and Comments. This section is borrowed from [EkeH1] with minor improvements. The idea of deriving information on the index of a critical point from the procedure by which it has been constructed is extremely important in modern nonlinear analysis. We will encounter it again in Sect. V.3.

4. Subharmonics

171

If Ker L = {a}, the normal form theorem reduces to the Morse lemma. The first proofs lost two orders of differentiability, so that Morse lemma held only for C 3 functions, until Cambini in [Cam] gave a C 2 proof. If Ker L = {a}, the normal form theorem was proved by Gromoll and Meyer in [GroM], assuming 'P to be C 3 • The improvement to C 2 was done by Hofer in [Hof2]. The statement we give here incorporates theorem 3 of [MawW] ch. 8.6 and the subsequent remark, which becomes condition (64). Mawhin and Willem provide a simple proof, following an idea of K.C. Chang [Cha].

4. Subharmonics We go back to the situation of Sect. 2, with a few changes. We shall assume that H(·, x) is T-periodic in t for every fixed x, and that H(t,·) is C 2 in x for every fixed t, with H' and H" continuous jointly in (t, x), and

(1)

H"(t,X)

positive definite for x

Theorem 2.1 tells us that if:

(2)

H(t,x) ~ H(t, 0) H(t,x) :s;

(3)

(4) (5)

H(t,x)

~

W

2"llxll

N"H(t,x)

2

for

for

lim sup H(t, x) O 0, a > 2 and w < 271' IT are suitable constants, then the Hamiltonian system

(6)

x=

JH'(t,x)

has a non-constant T-periodic solution. We note that H(·, x) is also kT-periodic in t for every k E lN, so that as long as

(7)

k

271'

0 so small that:

+ kek

(28)

Uk

(29)

Uk - hek E 'PI .

E

'Po

As we just noticed, ek is ~T-antiperiodic

(30)

ek

(t + ~T) +

ek(t) = 0 .

Since ('¢k)'Y = 'Po u 'PI, the origin belongs to 'Po or 'Pl. Say 0 E 'Po. Let Co be a path in 'Po connecting 0 to Uk + hek:

(31) (32)

'¢k (co(s))

< 'Y Vs E [0,1] .

Now act on Co by the phase shift s

-t S

by: (33)

CI(S)

:=

k

+ k[.

"2 * co(s)

This defines a new path CI

.

Using (30) and (31), we get (34)

CI(O)

=0

and

c(l)

= Uk

- hek .

The integral of a ~T-periodic function over one period does not depend on where we start integrating, that is:

(35) so '¢k (CI (s)) < 'Y for all s. So Uk - hek lies in the connected component of the origin in ('¢k) 'Y :

5. Autonomous Problems and Potential Wells

175

(36)

o

From (29) and (36) we get a contradiction. We now conclude the proof of Theorem 1. We claim that all the

Uk,

1 :S k :S N, are geometrically distinct. If not, there exist j < k and p E 7l. such that (37)

Xj =

p

* Xk

•

Write Uj = Xj and Uk = Xk, so that Uj = P * Uk. All the points in 7l. * Uk clearly have the same local properties, so we may assume that p = O. From now on, we write Uj = U = Uk and Xj = x = Xk. We have (38)

ikT(X)

(39)

ijT(X)

+ lIkT(X) :S1 :S ijT(X) + lIjT(X) •

:S1 :S

ikT(X)

On the other hand, using Lemma 1.4.9, we have: (40) If ikT(X) = 0, we get ijT(X) + lIjT(X) = 0 and contradict (39). If ijT(x) 2, we get ikT(X) ;?: 2 and contradict (38). So we must have:

+

lIjT(X) =

(41) By Proposition 2, we must have G(u) = k71.. But this contradicts the fact that U is jT-periodic, since j < k. Hence the result. 0 We could also ask whether G(Uk) = k71. for 1 :S k :S N, that is, whether has minimal period kT. This will not be true without further assumptions on H, which, however, turn out to be generic. It will also be true in the autonomous case, as we shall see in the next section. Xk

Notes and Comments. Take w = 0 (that is, H(t,x) = o(x2)) for the bake of simplicity. In [Rab3l, Rabinowitz proved the existence of a sequence nk -+ 00 of geometrically distinct subharmonics Xk, k E IN, with Xk having period nkT. The fact that we can take nk = k, that is, there exists a subharmonic of every possible period, is due to Ekeland and Hofer in [EkeH2].

5. Autonomous Problems and Potential Wells In this section, we turn to autonomous problems. We shall considerably extend the preliminary existence result of Sect. 2. Later on, we shall consider the particular case of second-order systems. Theorem 1. Let fl be a convex open subset of JR2n containing the origin. Let HE C 2 (fl, JR) be such that:

176

IV. Fixed-Period Problems: The Superlinear Case

'Ix ED,

(1)

(2) (3) (4)

'Ix 3w

-I- 0,

H" (x) is positive definite

> 0, 3€ > 0

H"(X)-l

H(x) :::;

0

-t

H(x) ~ H(O) = 0

W

"2llxll

2

Ilxll - t 00

when

for or x

Ilxll:::; €

-t

aD .

Then, for every T < ~, the system

(5)

{

:i; =

JH'(x)

x(O) = x(T)

o

has a solution with minimal period T.

Some comments are in order before we start the proof. By aD we denote the boundary of D; the theorem does not preclude the case when D = lR2n, so aD = 0. The precise meaning of assumption (4) is that for every a > 0, there exist p > 0 and rt > 0 such that: (6)

Ilxll

d(x,aD):::; rt => H"(x) ~ aI.

or

~ p

This implies that H(x) Ilxll- 2 - t 00 when Ilxll - t 00 or x - t aD. More precisely, for every a' > 0 there exist p' > 0 and rt' > 0 such that

(7)

IIxll

~ p'

or

d(x, aD) :::; rt' => H(x)

Ilxll- 2 ~ a'

.

Saying that x has minimal period T means that it is not T jk-periodic for any integer k ~ 2. As a consequence, it is not constant:

(8)

x(t)

-I- 0

'It.

We first prove the theorem in a particular case. Lemma 2. Assume that D = lR 2n, and H E C 2 (lR 2n, lR) satisfies conditions (1) to (4). Assume in addition that there exist 0: > 2 and r > 0 such that:

(9) (10)

lim sup H(x) Ilxll->CXl

Ilxll

~

r => H()"x) ~

2:,

Ilxll- a < 00 )..a H(x)

V)" ~ 1 .

Then, for every T < the system (5) has a solution x(t) with minimal period T, and we must have, for some to:

(11)

5. Autonomous Problems and Potential Wells

177

Proof. Once we have added assumptions (9) and (10), we are back in the situation of Theorem 2.1. We introduce the (reduced) dual action functional:

(12)

'¢(u)

:=

iT [~(JU,

IIu)

+ H*( -Ju)]

dt

on the space L~, and we find a critical point u = : by the AmbrosettiRabinowitz theorem. By Hofer's Corollary 1.13, we known that we can choose u to be either a mp - point or a local minimum .

In the first case, u, being a local minimum for '¢ on L~, will also be a local minimum for '¢OO on L':, and its index must be zero: (13)

iT(X) = 0 .

In the second case, by Theorem 3.9, the index must be zero or one: (14)

iT(X) ::; 1 .

We know that u i= 0; so x is non-constant and x(t) i= 0 for every t. The solution x is admissible; if it has period T /k, we have by Corollary 1.6.5: (15)

iT = i k.T / k

~

k - 1.

If iT(X) = 0, we get k = 1, and x has minimal period T. If iT (x) = 1, we get k ::; 2, and x has minimal period T or T/2. We have to show that it cannot have period T /2. We argue as in Proposition 4.2. Assume iT(X) = 1 and x is T/2-periodic. Then u is a mp-point. Denote by e i= 0 an eigenvector associated with the negative eigenvalue. By Bott's formula (Corollary 1.5.4) we have: (16)

iT = iT/2 + jT/2( -1)

while the theory of conjugate points (Corollary 1.6.5) yields (17)

iT

~

iT/2

+1 .

Since iT = 1, inequalities (16) and (17) imply that i T / 2 = 0 and ir /2 ( -1) = 1. This means that the eigenvector of QT associated with the negative eigenvalue belongs to the space (see Definition 1.5.3)

(18)

ET/2 := {Y E W 1,2 (0,T;JR 2n ) I y(O) +y(T/2)

=

O} .

Differentiating this eigenfunction, we get e, which must therefore be T /2antiperiodic: (19)

e(t + T /2)

=

-e(t) .

178

IV. Fixed-Period Problems: The Superlinear Case

By Theorem 3.9, 'ljJ'Y has exactly two components 'Po and 'PI, and we choose h > 0 so small that u + he E 'Po and u - he E 'Pl' Consider the 8 1 -action on L~ (phase shift): for 0 E 8 1

(20)

Define a continuous path c(O), starting at

c(O):= 0* (u+he) ,

(21)

We have 'ljJ (c(O)) = 'ljJ (c(O)) < This ends at:

"y

u + he,

.

by:

0:::; 0:::; 1/2

for every 0, so this path lies inside 'ljJ'Y.

(22) since u is T /2-periodic and e is T /2-antiperiodic. So we have connected u + he to u - he by a continuous path in 'ljJ'Y. This is a contradiction since they lie in different components, and our result is proved: x cannot have period T /2, so it has minimal period T in all cases. All that remains is estimate (11). Assume it does not hold, then: (23)

Vt,

H"

(x(t)) >

:;1

and equivalently:

Vt,

(24)

H" (X(t))-l
0 such that

H" (X(t))-l :::;

(~ - to) I

.

Consider the quadratic form on L~(O, T)

(26)

qT(V, v):= iT [(Jv, llv) +

(HII (u(t))-l Jv, JV)] dt .

The index iT(U) is the sum of the dimensions of the negative eigenspaces of qT (see Definition I.4.3). Inequality (23) yields

(27)

qT(V,V) :::; iT [(Jv,llV)

+ (~ - to) v2]

dt .

The operator J II has a 2n-dimensional eigenspace E1 associated with the eigenvalue !" (formula III.1.50). On this eigenspace, we have:

(28)

5. Autonomous Problems and Potential Wells

179

So iT(U) ;:::: dimEI = 2n. On the other hand, we have seen that iT(U) S l. We have a contradiction, and the lemma is proved. 0 We now reduce the general case to the particular case studied in Lemma

2. Proof of Theorem 1. Choose ho > 1 such that:

(29)

(x E Q

H(x);:::: ho )

and

=}

H"(x)

>

211" -T.J .

Set:

j'j

(30)

:= {x E QIH(x)

< ho }

j'j is an open bounded convex set. Construct a function ii E C 2 (IR2n, IR) such that

(31)

\/xE Q,

(32)

\/x

(33)

:lr

ii(x) = H(x)

rt. j'j ,

ii"(X)

Ilxll > r

> 0, :lc > 0 :

> 211" I T

=}

ii(x) =

C

IIxl14

One can, for instance, proceed as follows. Choose a number hI

> ho such

that

(34) Now consider the Fenchel conjugate H* of H. It is convex and finite everywhere. Choose some large number A > 0 and write

(35)

H"A(y) := H*(y)

if

Ilyll S

A,

+00 otherwise.

Take the Fenchel conjugate again, thereby defining HA := (H"A) * . It is convex, finite everywhere, and grows linearly at infinity. If A has been chosen large enough, we will have

(36) Now define

(37) For 8 > 0 small enough, we have

(38)

180

IV. Fixed-Period Problems: The Superlinear Case

Finally, choose an increasing C 2 convex function r.p : IR+ ~ IR+ such that

(39)

r.p(h) = h

(40)

r.p(h) = ,,/h2

for O:S h :S hI

for h 2: 2hI ,

where,,/ > 0 is a large constant. Now set (41)

H(x) := r.p (H1(x))

H is a convex function, which coincides with H on the set where H(x) :S hI, and with "/8 2 11x11 4 when IIxll is large. If "/ is large enough, we will have H"(x) 2: (~ + 1) I at every point x where H is C 2 and H(x) 2: hI. Smoothing H down, we get a C 2 function H satisfying (31) to (33). Apply Lemma 2 to H. We get a solution x of the problem (42)

± = JH'(x)

(43)

x(O) = x(T)

such that x has minimal period T and (44)

~"(_()) 21l" I . 3to : H x to :S T

Condition (29) then implies that the motion

H (x(t o )) < h o • Since H is an integral of

H (x(t)) < ho Vt. Condition (31) then implies that ii coincides with H on a neighbourhood of the trajectory x. SO H' (x(t)) = H' (x(t)), and x in fact solves the equation (45)

(46)

± = JH'(x) .

D

Theorem 1 does not apply directly to second-order system

(47)

q+v,(q)=o { q(O) = q(T) q(O) = q(t)

because the corresponding Hamiltonian (48)

H(p, q)

:=

1

2p2

+ V(q)

cannot satisfy assumption (4). A special statement is needed, with its own proof: Theorem 3. Let n be a convex open subset of IRn containing the origin. Let HE C 2 (n, IR) be such that

5. Autonomous Problems and Potential Wells

(49) (50)

Vq E R, Vq =f- 0 ,

3w

(52)

V"(q)-l

--+

Then, for every T
0, 3M : Vk ~ M , q ~ ilM ::::} V£'(q) ~ AI

Vk satisfies the additional conditions (56) and (57). By Step 1, the problem

{

(63)

q + V£(q) q(t + T)

= 0

=

q(T)

has a solution qk, with minimal period T and index:S 1. The latter is the index of the quadratic form 'l/J~(Uk) on L~, with Uk:= :ft22qk and

(64)

('l/J~(Uk)V,V) : =

iT [- (lIv)2

+ ((Vk)" (Uk)V,V)]

dt

= iT [-(lIv)2

+ (V£' (qk)-lv,v)]

dt.

The critieal point Uk for 'l/Jk is found by the Ambrosetti-Rabinowitz theorem (see Sect. 2). The corresponding critical value 'Yk is given by: (65)

'Yk

:= inf max

cET O~s~l

'l/Jk (c(s)) ,

°

where r is the set of all continuous paths c : [0,1] --+ L~/3 such that c(O) = and 'l/Jk (c(l)) < 0. Since 'l/Jk(O) = 0, we have 'Yk > 0, and since Vk :S Vk+1, we have Vk* ~ Vk*+l· It follows that the sequence 'Yk is bounded:

°

(66)

:S 'Yk+1 :S 'Yk .

By the duality Theorem IIA.2, we have: (67)

'Yk

=

iT

[~(p - Vk(qk)] dt .

5. Autonomous Problems and Potential Wells

183

We also introduce the constants: (68) If the Vk 0 qk, k E lN, are uniformly bounded, say Vk (qk(t)) :::; b, the problem is over. Indeed, it then follows from (60) that qk(t) E {h as soon as k ~ b, so that qk is in fact a solution of problem (47), with minimal period T. So all we have to do is to show that the sequence IlVk 0 qkll oo is bounded. Assume otherwise. Then we may assume that

(69) so that hk ---+ 00 by formula (68). We will now relate the hk to the 'Yk. We have, by adding (67)and (68): (70)

'Yk

+ Thk

=

iT q~dt

.

Take any M E lN, and set

(71) We take k so large that

[0, T] and set:

mk

> M. Denote by JL the Lebesgue measure on

(72) From Eqs. (68) and (70) it follows that:

(73)

'Yk

+ Thk ~

[ JVk(qk) M)::::} V£'(q)

~ 2 (~)

2

Pick the corresponding K from formula (75). With K fixed in this way, pick a vector ~ E IRn , and set, for t ~ 0:

184

IV. Fixed-Period Problems: The Superlinear Case

(78)

VK(t) := {

eJI-'K(t)~

if tEAK

o

otherwise .

The function VK has been built to have mean value zero, so VK E L~(O, T). Since AK is closed, [0, T] \AK is a countable union of subintervals (tn, tn + bn ), n E IN, so that: (79) Hence:

with: (81) We have:

(82)

t

(IIvK)' dt

~t

([

VK(S)ds)'

dt-~ (t ([ V'(S)d') dt)'

Substituting (79) and (81) into the right-hand side yields:

t

(83)

(IIvK)' dt

~ I'.(u)du

we see that (cp(T), x 0 cp) is a solution of (4). If E is a C 2 hypersurface, the equation ± = In(x) defines a flow on E, since In(x) is a C l field of tangent vectors, and the flow exists for all times since E is compact. Problem (4) consists of finding the closed trajectories of that flow. Definition 1. A solution (T, x) of problem (2) is called a closed characteristic on E. Two closed characteristics (T, x) and (S, y) are geometrically distinct if 0 there is no map cp : lRIT7l -+ RI S7l such that x = y 0 cpo

In other words, two closed characteristics are geometrically distinct if you cannot deduce one from the other by rescaling time. For instance, if (T, x) is a closed characteristic, so are

(T, Xl) ,

with

XI(t)

=

x(t + to),

for

to

(kT, X2) ,

with

X2(t)

=

x(t) ,

for

k E 1N

(Tla, X3)

with

X3(t)

=

x(at) ,

for

a> 0

E lR

but they are not geometrically distinct from x. The basic questions we want to answer are as follows. Are there any closed characteristics? If so, how many (geometrically distinct)? Can anything be said about their length and their (linear) stability? To cast the problem in Hamiltonian form, we proceed as in Sect. 1. 7 (Example 2), assuming that 0 E IntC. We first introduce the gauge jc : lR2n -+ [0, (0) of C

(6) (7)

jc(x) := Min

{>.I~ jc(O)

E C}

:= 0

for

xi- 0

1. Existence, Length, Stability

and then the Hamiltonian HOl : lR2n

--+

189

[0,00) defined by:

(8)

HOl is convex and positively homogeneous of degree Q. It is also C 1 if E is C 1 , and C 2 if E is C 2 and Q > 2. We refer to Example 1.7.2 for the Euler identities which hold in the C 1 case. If E is C 1 , then n( x) := H~ (x) clearly provides a continuous non-vanishing section of N E (x). With this choice of n( x), problem (4) becomes: HOl(X) = 1 { x = JH~(x) .

(9)

x(O) = x(S) More generally, we have: Lemma 2. Assume E bounds a convex compact set C with 0 E IntC. Set HOl = ib with Q > 1. Then any solution (TOl' x Ol ) of

HOl(X) = 1 { x E J{}HOl(x)

(10)

x(O) = x(S) is a closed characteristic of E. Conversely, if (T, x) is a closed characteristic of --+ lRITZ such that (TOl' x Ol ) with XOl = xocp,

E, there is a bijection cp : lRITOlZ is a solution of (10).

0

Proof. We have, for x E E:

{}HOl(x) = Q{}jc(x) = {x* E NE(x)l(x,x*) = Q}

(11)

so {}HOl(x) generates the convex cone NE(x). Since {}HOl C NE(x), any solution of (10) must be a closed characteristic of E. Conversely, if (T, x) is a closed characteristic of E, there is for every s some >.(s) > 0 such that x(s) E >'(S){}HOl(X(S)). The map x belongs to L1 (lRITZ;lR 2n ), and {}HOl , which has compact convex values, bounded away from the origin, is upper semi-continuous (that is, has closed graph). Using a measurable selection theorem, we find that we can choose the map>. : lRITZ --+ (0,00) to be L1 too. We then define a new time variable t = '¢(s) by

'¢(s) =

(12)

1 8

>'(u)du .

Since >.(u) > 0, the map '¢ : [0, S] --+ [0, TJ, with T := '¢(S), is a homeomorphism. Let cp be its inverse, and set XOl := x 0 cpo We have

dxOl

dt

(13) with s

=

=

dx dcp ds dt = x(s)>.(S)-l E {}HOl (x(s))

cp(t). So x solves problem (10).

o

190

V. Fixed-Energy Problems

As we already observed in Lemma 1. 7.2, the solutions (T, x) of the problem

{

(14)

X E JoHa(x) x(O) = x(S)

come in continuous families, indexed by the energy level h. To be precise, choose S = 1, and let Xl #- 0 be a solution of: {

(15)

X E JoHa(x)

x(O)

=

x(l)

Let h be its energy level:

(16) Then (To, xo) is a closed characteristic of E, with: (17) (18) This trick reduces the fixed-energy problem to the fixed-period problem. It gives us our first existence result.

Theorem 3. Let C C ffi2n be a convex compact set with non-empty interior. Its boundary E always carries a closed characteristic. Proof. The problem is translation-invariant, so we may assume that 0 E IutC. Choose a: E (1,2) and introduce the Hamiltonian Ha := je, where jc is the gauge of C. Consider the boundary-value problem:

(19)

{

X E JoHa(x)

x(O)

=

x(l)

By Corollary II1.3.6, it has a solution Xl with minimal period 1. In particular, Xl is non-zero, so it can be lifted to the energy level 1: the curve Xo given by formulas (17) and (18) is a closed characteristic on E. 0 Now that the question of existence is answered, we may start looking for qualitative properties of the closed characteristics. The first one is the length - the shorter the better. We shall give two basic estimates in the C l case. If x : ffilTZ - E is a closed characteristic, a natural measure of length in this context is the action A(T, x), defined by:

(20)

liT

A(T, X) := 2

(Jx, x) dt .

0

Indeed, if cp : ffil SZ - ffilTZ is any orientation-preserving diffeomorphism, we have A(T, X 0 cp) = A(T, x); so A(T, x) is a geometric quantity, depending only on how many times one runs around the closed characteristic.

1. Existence, Length, Stability

191

Denote by (Ta, xaJ the particular solution of problem (19) associated with the closed characteristic x. We have:

A(T, x)

(21)

fTa (Jx a , xa) dt ="21 fTa (Jxa,JH~(xa)) dt 1 fTa a fTa a ="2 (xa , H~(xa)) dt ="2 Ha(xa)dt = "2Ta 1

="2

0

0

0

0

where we have used the Euler identity on the function Ha, which is positively homogeneous of degree a. So 2a- 1 A(x) is the period of Xa. Taking into account the equation xa = JH~(xa) on H(xa) = 1, it is easy to deduce from the period a-I A(x) upper and lower bounds for the euclidian length of the trajectory of X a , that is, of the closed characteristic x.

Theorem 4 (Croke-Weinstein). Assume E is a C I hypersurface bounding a convex compact set C. Suppose C contains a ball of radius r > 0

(22) for some Xo E IR2n. Then, if (T, x) is a closed characteristic on E, we have:

(23)

A(T, x) 2: nr2 .

Proof. We shall work with a Xo = 0, so that:

=

o

2. Without loss of generality we may take

(24) Denoting by () the angle between x and H~ (x), we have

(25)

(x, H~(x))

=

IlxIIIIH~(x)11 cos(),

Vx E E

.

The tangent plane to E at x cannot intersect the ball Ilxll ::; r, since the latter is contained in C. It follows that Ilxll cos() 2: r, and hence:

(26)

(x, H~(x)) 2: r IIH~(x)ll,

Vx E E

.

By the Euler identity, the left-hand side is just 2, so that r IIH~(x)1I /2::; 1. Multiplying the right-hand side by this number, we get:

(27)

r2

(x, H~(s)) 2: 21IH~(x)1I

2

We now look at the solution (T2 , X2) of

(28)

{

X = JH~(x) x(O) = x(S)

, Vx E E

.

192

V. Fixed-Energy Problems

corresponding to the closed characteristic (T,x) by Lemma 2. We have X2(t) E E for every t, so that inequality (27) yields:

(29) Note that we can add a constant to X2 without affecting the left-hand side. Replace X2 by IIx2, the primitive with mean zero, and use Cauchy-Schwarz:

(30) By the Wirtinger inequality (Lemma 1.4.1), this yields (31) Since T2 = A(T,x), this is the desired result.

D

Proposition 5. Assume E is a C 1 hypersurface bounding a convex compact set C. Suppose C is contained in a ball of radius R > 0:

Vx E E,

(32) for some that

Xo

Ilx - xoll :::; R

E lR2n. Then, there is a closed characteristic (T, x) on E such

(33)

D

Proof. We shall work with 1 < a < 2. Without loss of generality, we may take = 0, so that

Xo

(34) Fix T > 0, and consider the (reduced) dual functional 'l/JT associated with the fixed-period problem (10):

'l/JT(U):= iT

D(Ju,IIu) + H~(-JU)]

dt

on L~ (0, T; lR2n) , with a-I + ,8-1 = 1. By Corollary 111.3.6, problem (14) has a solution XT such that UT := XT minimizes 'l/JT

(35) The left-hand side can be computed, by using the duality Theorem 11.4.2 and the homogeneity of HOI:

1. Existence, Length, Stability

'l/JT(UT) (36)

[~ [(JUT, lIuT) + H~ (-JUT )1] dt

=

iT

=

-iT

= iT

193

[~(JXT,XT) + Ha(xT)] dt

[~(H~(XT)' XT) -

Ha(xT)] dt

= -

(1- ~) THa(xT) .

We now have to adjust T =: Ta so that XT := Xa belongs to E. We thereby get a closed characteristic (Ta, xa) on E, and (36) yields (with Ua = xa)

(37) On the other hand, we know that Ha(x) ~

Ilxll a

R-a, and hence:

(38) This will enable us to estimate directly the right-hand side of formula (35). Indeed, setting

~a(U):= iTa [~(JU'lIU) + ~ lIull/3 (:a)/3- 1] dt

(39)

we have by (38):

(40)

Min 'l/JTa ~ Min ~a

.

We compute the right-hand side explicitly. It is given by formula (36), with He)! replaced by Ilxr R-a, namely: (41)

where (42)

xa

is a solution of: {

.

X

=

J

a

Ra

x

Ilxl12-a .

x(O) = x(Ta)

We remark that, by Corollary III.3.6, the solution period Ta. It is clearly of the form:

xa

must have minimal

(43) Adjusting w to fit the boundary-value problem (42) we get:

(44)

w = ;

Ilxa ll,,-2

194

v.

Fixed-Energy Problems

27r To:

(45)

W=-

and hence

(46) Writing this back into Eq. (41), we get:

(47) Hence, by inequality (40):

(48)

(1 - i) To: = > -

-1/lTa (uo:)

= -Min 1/lTo
c(.r) . -

Proof. We adapt the argument in Theorem III. I. 7 to a different setting, but the general trend remains the same. Recall that fJ is the geodesic distance in L~, defined by formulas (21) to (23) in Sect. IV.1, and that it is equivalent to the standard distance on bounded subsets. Choose E E (O,!) , and define Fe := {ulfJ(u, F) < E}

(7S)

Let A E :F be such that

(79) and consider the set r of all equivariant homotopies of the identity which leave A \ Fe := {u E Alu .;. Fe} fixed and are uniformly bounded from the identity. In other words, hE Co ([0,1] x L~; L~) belong to r if and only if:

v8,

(SO) (Sl)

h(8,U)=U

h( 8, . )

is equivariant

whenever 8=0 or uEA\Fe

supfJ(U,h(8,U))
lR by

(84)

I(h) := max{('1/1 + 'P)

0

h(l,u)lu E A}

-

EfJ(U, F)} .

where:

(85)

'P(u)

:=

max {O, E2

For every fixed u, the function h ---> ('1/1 + 'P) 0 h(l,u) is continuous on so I is lower semi-continuous. It is also bounded from below, since:

r,

2. Multiplicity in the Pinched Case

1(h) = sup {1jJ(v) ::::: sup {'¢(v)

(86)

::::: €2

209

+ .(l, u>.))

(111)

+