Chapter 14 Variational methods for hamiltonian systems

CHAPTER 14

Variational Methods for Hamiltonian Systems

EH. Rabinowitz* Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, W153706-1313, USA E-mail: rabinowi@math, wisc. edu

Contents 0. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part 1. Periodic solutions of (HS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I. 1. A technical f r a m e w o r k for periodic solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Superquadratic a u t o n o m o u s H a m i l t o n i a n systems . . . . . . . . . . . . . . . . . . . . . . . . . . .

1093 1095 1095 1097

1.3. Fixed energy results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1.4. Brake orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5. T i m e d e p e n d e n t superquadratic fixed period problems . . . . . . . . . . . . . . . . . . . . . . . . .

I i 01 1102

1.6. Perturbations from s y m m e t r y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7. Subquadratic H a m i l t o n i a n systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1103 1103

!.8. A s y m p t o t i c a l l y quadratic H a m i l t o n i a n s

1105

!.9. Singular potentials

.................................

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Part 2. Homoclinic and heteroclinic orbits

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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

II 09

2. I. T h e variational formulation of h o m o c l i n i c s to 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. S o m e results for h o m o c l i n i c s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I 110 IIII

2.3. S o m e basic heteroclinic results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. M u l t i b u m p solutions: the time d e p e n d e n t case . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I II 3 I117

2.5. M u l t i b u m p s in the a u t o n o m o u s case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3. Final remarks References

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*This research was supported by N S F grant # M C S 8 1 1 0 5 5 6 . U.S. G o v e r n m e n t is permitted. H A N D B O O K O F D Y N A M I C A L S Y S T E M S , VOL. I A Edited by B. Hasseiblatt and A. Katok 9 2002 Elsevier S c i e n c e B.V. All rights reserved 1091

Any reproduction

for the purposed of the

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O. Introduction During the past twenty years, there has been a great deal of progress in the use of methods from the calculus of variations to find periodic, homoclinic, heteroclinic, and other kinds of orbits for Hamiltonian systems. This is a very natural development. Indeed such solutions of Hamiltonian systems are global in time. Therefore it is reasonable to use global methods to obtain them in a direct fashion, working with a class of functions having the qualitative properties sought, rather than by means of approaches based on the initial value problem together with additional considerations. Moreover there are classical principles such as the Maupertuis Principle of Least Action and Hamilton's Principle that give a variational characterization of solutions of Hamiltonian systems. Indeed Birkhoff, Ljusternik and Schnirelmann, Morse, and others used these principles together with topological arguments and finite dimensional broken geodesic approximations to find geodesics. They were also used in minimization arguments to find periodic solutions. The variational formulation of general Hamiltonian systems in phase space, however, lead to functionals which are highly indefinite, e.g., their associated linearizations have an infinite Morse index. A lack of existence tools made it impossible to treat such functionals. However this obstacle was overcome through advances in critical point theory, especially as applied to the setting of semilinear elliptic partial differential equations. Indeed during the 1970s and 1980s, a collection of variational tools was developed and refined to the point that it could play a fruitful role in treating questions of the existence of periodic and subsequently more complicated orbits of Hamiltonian systems. The methods and classes of applications are still evolving so much more should be anticipated in the future from this area. The goal of this chapter is to present some variational formulations for problems of periodic solutions and connecting orbits, to survey some of the representative problems and results for Hamiltonian systems that have been obtained via variational methods, and to give an indication of some of the variational tools behind the results. No effort will be made to describe the most general or technically complicated settings. The interested reader can find more information in the papers cited below or in the useful monographs of Ambrosetti and Coti Zelati [9], Ekeland [60], Hofer and Zehnder [82], Mawhin and Wiilem [99] and the survey papers of Ambrosetti [4], Rabinowitz [ 111 ], and Zehnder [ 136]. Some additional material on variational methods can be found in Ambrosetti [3], Chang [47], Mawhin and Willem [99], Rabinowitz [106], Struwe [126] and Willem [135]. Turning to the equations that will be studied, let 17, q 6 R" and H 6 C I (I~ • It{2'', R). Then Hamilton's equations take the form

OH fi = - ~ ( t , p, q) -- - H q ( t , p, q), Oq OH [t -- ~ p (t, p , q ) - - Hp(t, p , q ) . Setting z = (p, q), the system becomes

~=JH:.(t,z),

J-

0 -id)

id

0

'

(HS)

P.H. Rabinowitz

1094

where id denotes the n • n identity matrix. Actually it is more natural to write (HS) as J~ = -Hr(t,z)

since this equation is the form the Euler equation takes, but we will stay with the more traditional (HS). As is well known, when (HS) is not forced, i.e., H = H ( z ) , H ( z ( t ) ) =constant for any solution so energy is conserved. The simplest special cases that arise in mechanics are H ( z ) -- K ( z ) + V ( q ) = kinetic + potential energy. For the simple case of K (z) -- 89Ipl 2 and V = V ( t , q) that will be treated often below, (HS) reduces to a second order system: + Vq (t, q) -- O.

(HS2)

This chapter will be divided into two main parts, each of which is subdivided into sections on more specialized topics. Part 1 deals with periodic solutions of (HS) and (HS2), and Part 2 concerns homoclinic and heteroclinic solutions. A brief section with some open questions follows Part 2. Before turning to Part 1, some of the questions treated in our survey will be mentioned. The two basic questions dealt with in Part l are (a) the existence of solutions of (HS) of prescribed energy, i.e., solutions of the autonomous system on a given energy surface, and (b) solutions of prescribed period for forced or autonomous systems. By means of technical devices, the best results for (a) have been obtained via (b). For both (a) and (b), there are further questions of the multiplicity of such solutions. When (HS) is forced periodically, e.g., with period T, a priori the forcing has period kT for all k E 1~1. Thus one can also seek so-called subharmonic solutions of (HS) which have period k T. In the unforced case for (b), solutions with prescribed minimal period can be sought. Growth assumptions on H(,z) as ]z.] ~ cx~ and the behavior of H as z, ~ 0 have also played an important role in studying the above existence questions. In particular, results often require some version of "superquadratic" growth: H ( z ) / I z l 2 ~ cx~ as Iz]--+ cxz or "subquadratic" growth: H ( z ) / I z l 2 ~ 0 as ]z,I ~ cx~. Another class of questions that arise for (b) concerns the effect of symmetries of H on the multiplicity of solutions. E.g., autonomous equations have a natural S I invariance in their variational formulation while if H is periodic in z l . . . . . z,,, the variational problem has a Z" symmetry. Such symmetries can lead to rich multiplicity statements beyond what the symmetry itself guarantees. A secondary question here is the effect of perturbations from symmetry on the multiplicity of solutions. A final question in Part 1 for (HS2) is the existence of periodic solutions when V is singular. Potentials with singularities arise in particular in celestial mechanics. Part 2 surveys variational work on homoclinic and heteroclinic solutions. There are two major subtopics here. The first is simply the existence of "basic" such solutions in various settings. The second topic is that of so-called multibump homoclinic and heteroclinic orbits. These are solutions of (HS) or (HS2) which are found near functions obtained by formally gluing or concatenating the basic solutions with appropriately chosen translates (in time) of themselves. This latter study is the variational analogue of classical results in the theory of dynamical systems involving symbolic dynamics and chaos. More will be said about the classical dynamical and current variational constructions in Part 2.

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Part 1. Periodic solutions of (HS) 1.1. A technical f r a m e w o r k f o r p e r i o d i c solutions To begin our survey of the variational approach to find periodic solutions of (HS), a technical framework will be introduced to treat such problems. As will be seen momentarily, from the point of view of functional analysis, a natural class of functions to work with here is the Sobolev space E =_ W ! / 2 ' 2 ( S I , II~2n). This is the space of 2n-tuples of 27r periodic functions which possess a derivative of order 1/2. Expressed in terms of Fourier series, if z E E, z = Z

a je ijr

j EZ where aj E C 2'' and a _ j -- a j. As norm in E, we have

Ilzll~; - Z ( l + .j EZ

I j l ) l a j l 2.

Write z E E as z = (p, q) with p and q being n-tuples and let p. q denote the inner product in R". For, e.g., continuously differentiable z, the action integral, A ( - ) is defined as 1 f~12x J z A ( z ) -- f~) 27r 17. gt dt -- -~

. " dt.

(1.1.1)

Viewing each of p and q as sharing the derivative that occurs in ( 1.1.1 ) shows why W I/2"2 is an appropriate function space here. Since IA (z)l <~ const IIz IIffc, A extends to E as a continuous bilinear tbrm which will continue to be denoted as in (1. I. l) for all z E E. It is not difficult to show that E has a splitting: E = E~

E + (9 E -

into subspaces E ~ E +, E - on which A is null, positive definite, and negative definite [106]. Moreover these subspaces are orthogonal in L 2 ( S j , R 2'') as well as with respect to the bilinear form B [ z , ~ l = fo rr (P " (P + gt " 99) dt

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PH. Rabinowitz

associated with A. Here ( -- (qg, ~P). Writing z 6 E as z = z ~ + z + + z - 6 E ~ 9 E + 9 E - , it is convenient to take an equivalent norm for E, Ilzll, where

]Z0]2 -']-A(z +) -

Ilzl12 =

A(z-).

In this norm, A (z) has the simpler form: A (z) -- IIz § I12 - IIz-112 making explicit the highly indefinite nature of A. For z 6 E and 1 ~< s < oo, standard e m b e d d i n g theorems imply

IIzlIL'I0,2~I-

(~o27r

[z(t) ]" dt

) l/s

~< ct.,. Ilzll

(1.1.2)

for some constant, o(, > O. Thus if H(t, z) satisfies a power growth condition for some s E [1,oo)"

IH(t, z)[ <~ ai + a2lzl"

(1.1.3)

for z 6 R 2'', the corresponding functional

7-[(-)----

f~)2rr H (t, z(t))

dt

(1.1.4)

can be shown to belong to C I (E, R) [ 1061. It will always be assumed that H satisfies (Hi) H 6 C I (I~ • R 2'', IR) with H T-periodic in t in the forced case. Note that H 6 C I does not guarantee unique solutions for the initial value problem for (HS) but this point is not an issue for the variational approach taken here. As was mentioned in the Introduction, results on periodic solutions of prescribed energy can be obtained from results on solutions of prescribed period. Thus the latter case will be studied first. By making a change of time scale, without loss of generality, H(t, z) is 2zr periodic in t. Consider the functional

l(z) -- A(z) - 7-[(z)

(1.1.5)

where H satisfies (1.1.3) so 1 6 C I ( E , R ) . l'(z) - - 0 , i.e., if ~" = (99, gr),

l ' ( z ) ( --

f0 rr[(ft.

gr +q9 . ~ ) -

Then critical points, z, of I in E satisfy

Hp(t,z).tp-

H q ( t , z ) . ~ ] dt = 0

(1.1.6)

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for all ~" e E. (To be more accurate, p and ~ , q and ~0 each share half of a derivative in the first term on the left in (1.1.6).) Thus, in the language of PDE, z e E is a weak solution of (HS) and an elementary regularity argument shows that z is actually a classical 2n" periodic solution of (HS). On the basis of these observations, the goal becomes that of giving conditions on H so that (1.1.5) has critical points in E and (HS) periodic solutions. The two main cases that have been treated involve (i) superquadratic and (ii) subquadratic growth for H ( t , z) as Izl~

~.

The next section treats the autonomous case for (i).

1.2. Superquadratic a u t o n o m o u s Hamiltonian systems The growth assumption required of H in this section is: (He) There are constants/z > 2 and r ) 0 such that 0 < # H ( z ) <~ z 9 H : ( z ) for Izl/> r. Condition (H2) implies a lower growth rate for H: there are constants a i, a2/> 0 such that for all z E R 2'' H (z)/> al Izl j' - a2. However there is no upper bound on the rate of growth of H. Suppose further that (H3) H ( z ) >~ 0 for z e •2,, and (H4)

H(z)=

o(Izl 2) as z ~

0.

Then we have THEOREM 1.2.1 ([ 1061). I f H sati,~fies (Hi)-(H4), then fi)r any T > 0, (HS) possesses a nonconstant T - p e r i o d i c solution. Qualitative and quantitative conditions like (HI)-(H4) lead to qualitative behavior for the corresponding functional in (1.1.5) that in turn can make abstract critical point theorems applicable. To illustrate briefly why this is the case in the current setting, for simplicity suppose T = 2rr. Unfortunately, H may not satisfy (1.1.3) and therefore 7-/(z) may not belong to C I (E, R). One gets around this difficulty by modifying H for large z so (1.1.3) holds and then getting appropriate a priori bounds for the modified problem. To avoid these technicalities, assume that (1.1.3) is satisfied so 7-/e C I (E, IR). By (H3), I <~ 0 on E ~ E while by (H4), I (z) >//ilzll 2 for small z e E +. Using (H2)-(H3), a set of the form Q = {re + 10 ~< r ~< rl} G {z e E ~ G E - I Ilzll ~ r2 }

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P.H. Rabinowitz

can be constructed (where e + 6 E + and Ile§

- 1) such that l]i}Q ~ O. Topologically, Q

links a small sphere in E +. Furthermore I satisfies the Palais-Smale condition, a useful compactness condition henceforth denoted by (PS) which says whenever I (urn) is bounded and 1'(99,1) --+ 0 as m --+ e~, then (urn) possesses a convergent subsequence. A version of the Mountain Pass Theorem [ 106] now implies I has a critical point z with I (z) > 0. Since I ~< 0 on E ~ z is a nonconstant solution of (HS). REMARK 1.2.2. Although Theorem 1.2.1 guarantees that (HS) has a nonconstant Tperiodic solution for any T > 0, it says nothing about whether the solution has minimal period T. After some initial progress on this question by Ambrosetti and Mancini [ 10] and by Girardi and Matseu [71 ], Ekeland and Hofer [61] showed THEOREM 1.2.3. If H ~ C2(II~ 2'', IR), satisfies (H2)-(H4), and H is strictly convex, then for any T > 0, (HS) has a T-periodic solution with minimal period T. To prove Theorem 1.2.3, Ekeland and Hofer used the convexity of H to invoke the socalled dual variational method to transform (HS) to a new variational problem to which the usual Mountain Pass Theorem applies. Additional index theoretic computations involving critical points of mountain pass type then yield a solution of minimal period T. REMARK 1.2.4. The dual variational method employed in Theorem 1.2.3 is a useful tool for problems where strict convexity is present. It is essentially an infinite dimensional Legendre transformation. Writing (HS) as -,.7~. -- H : ( - ) ,

(1.2.5)

the strict convexity of H can be used to invert H: obtaining z = H---~ (-,.7~.). ,,.

(1.2.6)

Taking " as a new independent variable, (1.2.6) can be reformulated as a variational problem. This approach was initiated by Clarke [51,52]. Returning to the setting of Theorem 1.2.3, several authors have obtained minimal period results under milder conditions than convexity, especially for (HS2). See, e.g., [72,91,90, 92]. For another aspect of Theorem 1.2.1, observe that if z ~ E and 0 ~ IR,

l(z(t-0))-

l(z).

(1.2.7)

Thus due to the 2n" periodicity of z, I has a natural N/[0, 2 7 r ] - S I symmetry. Indeed such a symmetry is present whenever one seeks periodic solutions of an autonomous Hamiltonian system. No use of this symmetry was made in Theorem 1.2.1. However it can be exploited to get a much stronger result. In fact topological index theories such as Ljusternik-Schnirelmann category have been developed which measure the topological complexity of symmetric sets. Assumptions of a group symmetry, e.g., S 1 or Z", in the

Variational methods for Hamiltonian systems

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setting of various abstract critical point theorems then lead to much stronger statements on the number of critical points the functional possesses. As applied to the superquadratic setting of Theorem 1.2.1, these tools yield" THEOREM 1.2.8 ([105]). If H satisfies (HI)-(H2), then f o r any T, R > 0, (HS) has a T periodic solution, z, with IIZlIL~IO,TI ---- maxtclO, Tl Iz(t)l /> R. Counterexamples in [104] show without a hypothesis like (H4), one cannot expect solutions having an arbitrary minimal period.

1.3. Fixed energy results As was mentioned in the Introduction, results for (HS) when the period is prescribed can lead to results when the energy is given. To see why this is so, suppose, e.g., H - I ( 1 ) is a compact hypersurface in It{2'1 and we seek periodic solutions of (HS) on H -I (I). It will be shown how to find such solutions via Theorem 1.2.1 for the special case of H - I ( 1) bounding a star-shaped neighborhood of 0. See, e.g., [ 106]. Another approach can be found in Weinstein [133]. For such an energy surface, each z 6 II{2"\{0} can be written uniquely as z = o t ( z ) w ( z ) where w ( z ) ~ H - I ( l ) and ct(z) --I=l/Iw(-)l. Introduce a new Hamiltonian" H(z) --ct(z) 4. Then H(z) 6 C I (•2"\{0}, R), is homogeneous of degree 4, and H--I ( 1) -- H - I ( 1). If a nonzero 27r periodic solution of 9

,TH--~(:)

(I.3. l ) m

can be found, by conservation of energy, H (z(t)) =-- constant -- c > 0. Moreover if ?, is a constant, ~"(t) = y z (t) satisfies _ y-2j/~_(()

(1.3.2)

and , q ( ( ) }/4C. Thus choosing y = c - I / 4 shows ( lies on H - I ( l ) and is a periodic solution of (HS). Finally the existence of a 2rr periodic solution of ( 1.3.1 ) follows since H satisfies (Hi)-(H4) of Theorem 1.2.1. Based on the existence result for this star-shaped case and some related results, Weinstein conjectured [134]" -

-

THEOREM 1.3.3. I f H - I ( I ) is a c o m p a c t manifold o f contact type with H l ( S , II{) = 0 , then (HS) has a periodic orbit on H - I ( l ). This conjecture was proved by Viterbo [13 l]. Subsequently Hofer and Zehnder [81] simplified Viterbo's proof and further showed: THEOREM 1.3.4. Suppose H ~ CI(II{2",R) a n d H - I ( 1 ) compact region. Then there is a sequence em ~ era) contains a periodic orbit z,, o f (HS).

0 as m ~

is a manifold bounding a oo with em =/: 0 and H - I ( l +

1 1O0

P.H. Rabinowitz

The proof of Theorem 1.3.4 is in the spirit of the star-shaped case although it is much more complicated. It involves a reduction to a fixed period case and use of a generalized Mountain Pass Theorem. By Theorem 1.3.4, whenever one can find a priori m-independent upper and (positive) lower bounds for the period, T,,,, of z,,,, an elementary limit argument shows that (HS) has a periodic solution on H - i ( I ) . The bounds on T,,, can be obtained from upper and lower bounds on the action integral, A(zm). E.g., it was shown in [20] that such bounds obtain whenever one has a "classical" Hamiltonian system" H ( p , q ) = /jpj2 A-- V ( q ) where V 6 C i (R", R) and V - I ( I ) is a compact manifold. Direct existence proofs for this classical case prior to the discovery of Theorem 1.3.4 were given by several authors" Bolotin [31 ], Gluck and Ziller [73], Hayashi [76], and Benci [ 19]. In fact they found brake orbits, a topic that will be discussed in Section 1.4. More general sufficient conditions for bounds on T,,, can be found in Hofer and Zehnder [82]. An open question that received much attention was whether such bounds could be obtained for a general H with, e.g., H - i ( I ) diffeomorphic to S 2''-I. Counterexamples have now been obtained for n ~> 4 by Ginzburg [69] and by Herman [78] for n ~> 3. The case of n -- 2 remains open. A more direct approach to the fixed energy problem is to attempt to find a critical point of A on the set 7-/-I (27r). If all goes well and this constrained variational problem has a solution, the resulting Euler equation is (I.3.5)

9 =XJH:(-)

with X as Lagrange multiplier. Moreover since (1.3.5) is a Hamiltonian system, H (-(t)) = constant - c so ~(z)-

2 7 r ( ' - 27r

(1.3.6)

and c = 1. Thus H ( z ( t ) ) = I so z lies on H - I ( l ) . Finally rescaling time, z is a 27r/X periodic solution of (HS). Although this method can be carried out in some interesting cases, there are several technical difficulties in making it work. The set ~ - I (27r) may not be a nice manifold, an appropriate variational principle may not be easily obtainable, and the Palais-Smale condition may not be satisfied. To conclude this section, a few remarks on the multiplicity of periodic orbits on H - I ( 1) are in order. The first significant result of this nature is due to Ekeland and Lasry [64] who showed THEOREM 1.3.7. Let H ~ CI(R2", R) with H - I ( l ) a manifi)ld which bounds a strictly convex neighborhood of O. Set R--

max ~!!

t(I)

1~'1 and

r--

min

[~'[.

~'ctt-I (I)

tf R

1 < - - < x/2, r

then H - I ( 1) contains n geometrically distinct solutions of (HS).

(1.3.8)

Variational methods for Hamiltonian systems

I l 01

The proof of this theorem involves using the S ! symmetry of the associated functional and some comparison arguments that exploit the pinching condition (1.3.8). Some generalizations of Theorem 1.3.7 have been obtained. E.g., Berestycki, Lasry, Mancini, and Ruf [21 ] proved a related result for H - I (1) nested between a pair of ellipsoids. This latter result can be used to give a new proof of a bifurcation theorem of Weinstein [ 132] which in turn is a generalization of the Lyapunov Center Theorem. Perhaps the major open question on periodic solutions of (HS) is whether the pinching condition (1.3.8) is necessary for the conclusion of Theorem 1.3.7. A partial answer was given in work by Ekeland and Lassoued [65], Szulkin [127], and Ekeland and Hofer [62] which yields: THEOREM 1.3.9. I f n >~ 2, H E CI(IR2's,IR), and H - I ( l ) is a manifold b o u n d i n g a c o m p a c t strictly convex region, then there are at least two distinct periodic orbits o f (HS) on H - I (1). For the case of n = 2, Hofer, Wysocki, and Zehnder recently [80] have obtained a very strong result: THEOREM 1.3.10. I f H ~ C2(IR 4, IR) and H - l ( l ) is a man~fold bounding a c o m p a c t strictly convex region, there are either two or it!finitely m a n y periodic orbits o f (HS) on H-I(l). An example of the first alternative is given by

H(zl, z2) =

Izll2 "~

r i-

Iz212 "3

r2

where rF/r~- is irrational and there are exactly two periodic solutions of (HS) on the ellipsoid H - I (1). Further interesting results for the fixed energy case can be found in the book of Ekeland [60].

1.4. Brake orbits The first existence results on periodic solutions of (HS) of given energy are for so-called brake orbits. Such orbits are most easily described for (HS2). Suppose V -I ( 1) is a compact manifold bounding a region D in R". A brake orbit of (HS2) is a solution of the equation such that q(0) 6 V - I ( I ) , for some T > 0, q ( t ) ~ D for t 6 (0, T), and q ( T ) E V - I ( l ) . Conservation of energy shows ~,(0) = 0 = Q(T). Hence q can be extended as an even function of t about 0 and T resulting in a 2T periodic solution of (HS2). A similar device can be used when H ( p , q ) is even in p [133]. In a seminal work that wasn't improved on for 30 years, Seifert [ 118] used geodesic arguments from geometry to find brake orbits for a class of Hamiltonian systems. He assumed

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H (p, q) -- ~ aij (q) pi Pj + V (q) i,j= 1

(1.4.1)

with (aij(q)) positive definite for q 6 79 and 79 diffeomorphic to the unit ball in R n. Subsequently Weinstein [133] went beyond Seifert's work employing new geometrical arguments to show if H (p, q) is even in p and H -j (1) bounds a convex region in II~2", then H - I (1) contains a brake orbit. At the same time, Bolotin [31 ] treated H as in (1.4.1) with V-! (1) a manifold bounding a compact region and obtained a brake orbit on H-~ ( 1). This result was later obtained independently using different arguments by Gluck and Ziller [73], Hayashi [76], and Benci [19]. There are also some results on the multiplicity of brake orbits in the spirit of Theorem 1.3.7. See, e.g., van Groesen [130] and Ambrosetti, Benci and Long [6].

1.5. Time dependent superquadratic fixed period problems Thus far only autonomous problems have been treated for (HS). The case where H also depends on t will now be considered. As was mentioned in the Introduction, one can now also seek subharmonic solutions. The main type of result here is an analogue of Theorem 1.2.1: THEOREM 1.5.1 ([ 104]). Suppose H sati.~fies (HI)-(H4) (H also being T-periodic in t); and (Hs) there are constants ot > O, [4 >~0 such that for Izl ~ [4,

]H:(t, z) I <~~ ' . H:(t, "). Then for each k e 1~, there is a k T periodic solution of (HS). Moreover infinitely many of these solutions are distinct. The existence of the zk's follows from a version of the Mountain Pass Theorem. That infinitely many are distinct is a consequence of estimates for the corresponding critical values and a comparison argument. Technically hypothesis (Hs) is needed to replace the energy conservation that is available in the autonomous case. In this connection, it is interesting to observe that for (HS2), the analogous result to Theorem 1.5.1 does not require more than the appropriate versions of (Hi)-(H4) [ 104]. This turns out to be the case because of better embedding theorems for W !'2 than for W 1/2'2. In particular []q IIg ~10, 7"1 ~< const. [[qllw~21o,Ti while this estimate does not hold with W 1'2 replaced by W I/2"2. Variants of Theorem 1.5.1 have been obtained by several authors. E.g., Ekeland and Hofer [63] replace (Hs) by an assumption of strict convexity for H and find a family of z,, all of which are distinct. See also [60,99], and [82].

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1.6. Perturbations from symmetry A special class of problems of the type of Section 1.5 arises in the setting of Theorem 1.2.8. In Theorem 1.2.8, exploiting the S l symmetry of the functional, I, an unbounded sequence (in L ~ ) of critical points was obtained. One can therefore ask what happens if a time dependent perturbation is made which destroys the symmetry. As the simplest example, consider

-- J ( H = ( z ) + f (t))

(1.6.1)

where H is as earlier and f is continuous and T-periodic. This kind of problem was studied by several authors beginning with Bahri and Berestycki [ 14] who proved" THEOREM 1.6.2. Suppose H ~ C2(R 2'', R) and f ~ C(It~, R 2'') is T-periodic in t. If H satisfies (H2) and (H6) H ( z ) <~ a3 + a4lzl s f o r z ~_ I~2'' where s < 2lz, then (HS) possesses an unbounded sequence of 7"-periodic" solutions.

The proof of Theorem 1.6.2 involves a mixture of analytical, topological and variational arguments, both minimax and Morse theoretic. The hypothesis (He,) is needed for certain comparison arguments that play a role in the proof. It is an interesting open question as to whether it is needed. Indeed for the analogous situation for (HS2):

;~ + V'(q) - f ( t )

(1.6.3)

which was also treated by Bahri and Berestycki [ 15], no condition like (H6) is required due to better comparison arguments than are available tbr ( 1.6.1 ). There have been several generalizations and improvements on [14,151. See, e.g., Long [88,89,87]. A recent paper of Bolle, Ghoussoub, and Tehrani [30] also treats some boundary value problems related to ( 1.6.1 ) and (1.6.3).

1.7. Subquadratic Hamiltonian systems There are many results on periodic solutions for subquadratic Hamiltonian systems. A representative sample will be described in this section. From the point of view of mechanics, perhaps the most interesting and important subclass occurs when (H7) H is T-periodic in t and 1] periodic in Zi, 1 ~< i ~< 2n. When (H7) holds, if z is a periodic solution of (HS), so is z + (kl TI . . . . . k2,, T2,,) for all k -- (kl . . . . . k2,,) ~ Z 2''. Consequently periodic solutions of (HS) occur in such Z 2'' equivariant families. Alternatively (HS) can be viewed on the 2n-torus. A multiplicity

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P.H. Rabinowitz

theorem for (HS) in this setting was obtained in a celebrated result of Conley and Zehnder [54], settling a conjecture of Arnold: THEOREM 1.7.1. Suppose H ~ C2(R • R 2'' , ~ ) and satisfies (H7). Then (HS) has at least 2n + 1 distinct families o f periodic solutions (and at least 22'1 if they are all nondegenerate critical points o f the associated functional). In fact, viewed on T 2n, these periodic solutions are all contractible. See, e.g., the book of Hofer and Zehnder [82] for more information and related results. The proof of the theorem given in [54] uses a Lyapunov-Schmidt reduction to a finite dimensional problem where ultimately the fact that catT/TJ = j + 1 is used. Here catx Y denotes the LjusternikSchnirelmann category of the set Y relative to X D Y. Other authors have found different proofs as well as generalizations to the case where 7-/is replaced by Q(z) + H where Q is quadratic in z. These new proofs also play back to the same basic topological facts. See, e.g., Chang [46] and Mahwin and Willem [99]. In a rather different setting, Clarke and Ekeland [53] studied a family of subquadratic Hamiltonians where H satisfies (Hg) H ( z ) / I z l 2 ---> ~

as Izl--> 0

and (H,~) H ( z ) / I z l 2 ---> 0 as I=1 ~ ~ . They used the dual variational method to obtain the first variational results on solutions having a prescribed minimal period. THEOREM 1.7.2. Suppose H sati,sfies (Hi), (Hs)-(H,~) with H convex, H ( 0 ) = 0 , and 11=.(0) = O. Then f o r all T > 0, (HS) has a T-periodic solution with minimal period T. A comparison argument was used to show the critical points they find have minimal period T. If condition (Hs) is dropped, there may not exist solutions with minimal period T for all T > 0. A counterexample can be found in [99]. Several authors have shown however that if (Hg) is strengthened, there are nonconstant periodic solutions for all large T without (Hs). A typical such result is the following due to Mahwin and Willem [99] which generalizes work of Brezis and Coron [421. THEOREM 1.7.3. If H is convex and satLsfies (Hi), (H,~), and (HI()) H ( z ) --> ocz as [z[--+ oo, then there is a T() > 0 such that f o r all T > 7]), (HS) has a periodic solution, z, with minimal period T. Moreover mint~l(),TI Iz(t)l --+ 0 as T ---->~ .

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The proof of Theorem 1.7.3 involves a generalized Mountain Pass Theorem and comparison arguments. Related results have also been found for forced Hamiltonian systems. The following due to Mawhin and Willem [99] is a good representative of this class: THEOREM 1.7.4. I f H is convex in z and satisfies (HI), (H9)-(HI0), the latter uniformly in t, then f o r each k 9 1%I,there is a k T periodic solution o f (HS). M o r e o v e r as k --+ cxz, ]]Zk ]]t.~ ~ cx~ and the minimal p e r i o d o f zk tends to cx~ as k ~ cx~. For nonconvex Hamiltonians, there are a few results based on (Hi0) and further conditions on H. E.g., using variational linking arguments, Silva [ 123,124] proved: THEOREM 1.7.5. I f H satisfies (Hi), (Hll) there is an M > 0 such that In~(t, z)l ~ M f o r a l l t 9 [0, T] a n d z 9 •2,,, a n d (Hi0) holds f o r H or - H uniformly f o r t E [0, T], then f o r each k 9 1~, (HS) has a k T periodic solution, Zk, a n d []Zk lit.~ -+ ~ as k --+ cx~.

It is interesting to contrast Theorem 1.7.5 with the following result, also due to Silva [123,124] and also based on variational linking arguments: THEOREM 1.7.6. Suppose H satisfies (HI), H ~ 0, (HI2) H ( t , z), IHr(t, z)l ~ 0 as Izl ~ ~ uniformlyfi~r t E [0, T], and

(HI3) H ( t , z) <~ 0 (or ~ O)for all t E [0, T] and z E ~2,,. Then there is a sequence (k i) C I~ with k.i --+ cxz as j ~

~

a n d distinct kj T-periodic

solutions z i o f (HS).

Earlier versions of the last two results are due to Giannoni [70] who used a critical point theorem, the so-called Saddle Point Theorem, to prove them.

1.8. Asymptotically quadratic Hamiltonians Several authors have studied a class of problems that stands between the super- and subquadratic cases described so far. They assume H(t, z) = Q(z) + R(t, z) where R(t, z) = o(Izl 2) as Izl ~ ~ and get existence and multiplicity results for (HS). Clark [50] studied such a problem for (HS2) with V even in q. The question for (HS) was initiated by Amann and Zehnder [11,12]. See, e.g., [99] for more on such results.

P H. Rabinowitz

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1.9. Singular potentials Aside from some early work of Gordon - see, e.g., [74], applications of modern variational methods to singular Hamiltonian systems began in the late 1980s with papers of Ambrosetti and Coti Zelati [9] and Greco [75]. There has been a great deal of subsequent work on this topic. Most of it has been done in the setting of (HS2) so we restrict ourselves to that case. For a more complete description of results on singular potentials, the monograph of Ambrosetti and Coti Zelati [9] and survey paper of Ambrosetti [4] can be consulted. Hamiltonian systems with singular potentials, i.e., potentials that become infinite at a point or a larger subset of R" arise in celestial mechanics. E.g., the Kepler problem with V ( q ) = - l / I q l has a point singularity at 0 while the n-body problem with V -V(qt . . . . . q,,) with qi E R e contains terms of the form ]qi - qjl -I for i :/: j and has a more complicated singular set. To formulate such problems variationally, consider the simple case of V (t, q) where V satisfies (Vl) V 6 C l ( R x ( R " \ { 0 } , R ) (V2) V ( t , x ) ~ - ~

andVisT-periodicint.

as x ~ O,

and

( V 3 ) V (t, q ) < 0 for q --/:0. In seeking T-periodic solutions of (HS2), the associated functional is

I(q)-

L"t',

-~1il

- v(t,q)

)

dr

(1.9.1) ] ")

and because of (V2), I is not defined for all q in the Sobolev space W,r -([0, T], R"), the closure of the space of smooth T-periodic functions under LT

2)

(1012 + Iql

Ilqll 2 -

dt.

This difficulty can be eliminated by requiring that q belongs to the Hilbert manifold, A defined via I 2

A = {q E W,r" ([0, TI

,Rn

],9

Indeed since q E W.r - implies

IIqllL~tO,7t ~ const IIqll,

.

) I q(t) =/=0 for all t E [0, T]}

.

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1107

it can be assumed that q is continuous and therefore the pointwise condition q(t) ~ 0 makes sense. However there still remains a problem in dealing with I on A due to the behavior of I near 0 A. This difficulty has been avoided by primarily studying potentials which satisfy the so-called strong force condition introduced by Gordon [74]:

(V4)

There is a neighborhood, A/', of 0 and U e C I (.Af\{0}, It~) such that IU(x)l ~ as x --+ 0 and

]U'{x}[ 2 <~-v{t,x} for x E A/'\{0} and t e I1~. A solution of (HS2) passing through the singularity, 0, of V is called a collision orbit. A consequence of (V4) is that (HS2) has no periodic collision orbits in W~.'2 with I (q) < oo. In fact if q,,, ~ OA as m --+ ~ ,

-

f0 "

V (t, q,,, ) dt ~

~.

To study the singular case further, suppose that as in the Kepler problem, (Vs) V ( t , q ) , IWq(t,q)l ~ 0 a s Iql ~ ~ , uniformly for t E [0, T]. Ifqm ------Rm e IK't and IRml ~ ~ as m ~ c~, then l(qm) --+ 0 and l'(qm) ~ 0. Thus (qm) is a Palais-Smale sequence with no convergent subsequence. Hence the (PS) condition fails here. However to apply variational methods one only needs a local version, (PS),, of (PS). Namely is suffices that l(qm) ~ c and l'(qm) ~ 0 as m ~ ~ implies (q,,,) has a convergent subsequence. This should hold for values of c which one hopes to show are critical values of I. The example of q,,, -- R,,, implies that (PShl fails. But under ( V I ) (Vs), it can be shown that (PS), holds for all ~, > 0. For this setting, Ambrosetti and Coti Zelati have proved [9]: THEOREM 1 . 9 . 2 . values.

If V sati,sfies (Vi)-(Vs), then I has an unbounded sequence ~?f critical

More generally, (V3) and (Vs) can be replaced by (V~,) There are constants p, M > 0 such that

Iw~t,x)l + Iw,,~t,x)l ~ M

for Ixl/11

and the same conclusion obtains. Moreover (HS2) possesses infinitely many distinct subharmonic solutions [108]. See also Majer [93]. The proofs of these results involve minimax arguments in the spirit of Ljusternik and Schnirelmann together with results

P.H. Rabinowitz

1108

of Fadell and Husseini [66,67] which establish that the set of unbased loops has infinite category. In "classical" celestial mechanics problems, the strong force condition (V4) generally is not satisfied. However when the remaining above conditions (VI)-(V3), (Vs) hold, one can perturb (1.9.1) by a term which makes Theorem 1.9.2 applicable, e.g.,

Is(q) = l(q) + 3 ~ T Iq1-4 dt. Moreover one can obtain 3-independent bounds on the critical values of Theorem 1.9.2 and corresponding 6-independent bounds for the critical points. Hence letting 3 --+ 0 yields what might be called a generalized or weak solution of (HS2). Such generalized solutions lie in Wr]:2 vanish at worst on a closed set, 79, of measure 0, are in C2([0, T]\79) and satisfy (HS2) on [0, T]\79. See, e.g., [17]. It is then an interesting but little studied regularity question as to whether these solutions are actually classical solutions or are collision orbits. Under some further hypotheses, a few authors have used Morse theory to help obtain bounds on the number of possible collisions and in particular have shown that there are no collisions in some special cases. See, e.g., Coti Zelati and Serra [58] and Tanaka [129]. Singular systems which don't satisfy (V4) are sometimes called weak force systems. Degiovanni and Giannoni have studied some such examples, including potentials like V(q) ~ - I q l -I . They have established the existence of noncollision periodic orbits with the aid of comparison arguments. See, e.g., [59]. Another class of singular problems that have been studied involve potentials of n body type where .vl . . . . . x,, E I1~~, x = (.,el . . . . . . '0, ) 1

It

v (r, x ) - -~ ~

v~j(t, x~ - x j)

(1.9.3)

i#j=l and Vii(t, z) are T-periodic in t, < 0, and -+ - o o as z ~ 0. Here Coti Zelati [55] showed: THEOREM 1.9.4. V is autonomous and of the form (1.9.3) with Vii sati,s;~,ing (Vl)-(V4)

(n being replaced by g) and (V7) gi.j(x)--- gji(x),

Then fi~r each T > O, the associated Hamiltonian system has infinitely many T-periodic solutions. If (V4) is dropped, there exists a generalized T-periodic solution. Theorem 1.9.4 is proved by minimizing the associated functional

l(q) --

Z I~,i12- V(t, q)) dt

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1109

1,2 over the class of functions ql . . . . . q,, 6 WT ([0, T], R e) such that qi(t) :/: qj(t) if i :/: j for all t 6 R and

q t +-~

---q(t).

(1.9.5)

Thus a family of symmetric solutions in the sense of (1.9.5) is obtained. More elaborate variational arguments were employed by Bahri and Rabinowitz [ 18] to get periodic solutions for strong force potentials of 3-body type and also generalized solutions when the strong force condition was dropped. Riahi [ 116] extended these results to the n-body setting. Some other who have used interesting minimax arguments on n-body type strong force potentials to get solutions of prescribed period or energy are Majer and Terracini [94,95], Serra and Terracini [122], and Ambrosetti and Coti Zelati [9]. A major open question here is to get regularity results for the weak force case.

Part 2. Homoclinic and heteroclinic orbits The use of variational methods to find connecting orbits, in particular homoclinic and heteroclinic orbits, of Hamiltonian systems begins with the work of Morse [103] and Hedlund [77] on heteroclinic geodesics in the late 1920s and early 1930s. Subsequently Kozlov and Bolotin found heteroclinics for multiple pendulum type problems. See, e.g., Kozlov [85], and Bolotin and Kozlov [37]. Building on the experience gained in the search for periodic solutions, activity in this area grew in the late 1980s and has been increasing since then. The technical difficulties encountered in seeking connecting orbits go beyond those of the periodic setting in three major ways. First there is a loss of compactness in treating problems on spaces of functions defined on the unbounded intervals R or R + rather than on S I as in the periodic case. This makes the study of (PS) sequences more complicated here. Indeed the (PS) condition is generally not satisfied for connecting orbit problems. However it is precisely in understanding how the (PS) condition breaks down that leads to existence results, especially for multibump solutions that will be described below. Fortunately work on problems in geometry and PDE paved the way for the study of the (PS) sequences encountered here. The seminal work of Sachs and Uhlenbeck on the YangMills equations [117], Bahri and Coron on limit exponent problems for semilinear elliptic PDE's [16], EL. Lions on concentration compactness [86], and Bahri on critical points at infinity [13] led to the analysis of the (PS) sequences here in a precise analytical way. A second difficulty encountered for connecting orbit problems is that although there may be a natural class of curves or functions to work with, there is not always a clear choice of an associated norm or metric. Finding a good setting in which to formulate the variational problem is often a basic difficulty. Finally the functionals associated with a given connecting orbit problem may be infinite on the natural class of curves or functions. Thus one has to find a "renormalized" functional and once again there is no general recipe for doing this although there have been a few successes.

PH. Rabinowitz

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A few results for connecting orbits have been obtained in the setting of (HS), beginning with Coti Zelati, Ekeland, and $6r6 [56], $6r6 [119,120], Hofer and Wysocki [79], and Tanaka [128] for superquadratic Hamiltonians, and by Felmer [68] and Cieliebak and $6r6 [48] for settings like Theorem 1.7.1. However most of the research on homoclinics and heteroclinics has been carried out for (H~2) so our survey will focus on that case. Two main topics will be treated in this setting. First existence results for basic homoclinic or hete~'oclinic solutions of (HS2) will be described. Then more complicated solutions, the so-called multibump solutions, that are found variationally near formal sums of basic solutions will be discussed. Here a formal sum means an actual sum if we are dealing with homoclinics to 0 or what is obtained by concatenating or gluing basic solutions for the case of heteroclinics.

2.1. The variational formulation of homoclinics to 0 The simplest case of connecting orbits to consider is that of solutions of (HS2) that are homoclinic to a point. E.g., Let E = W 1.2 (R, IR") with norm iiq]]2 _ ~ ( i ~ ] 2 + [q[2) dt.

Set L(q) -- ~lql 2 - V(t, q), the Lagrangian for (HS2) and

l(q)--JllL(q)dt.

(2.1.1)

If, e.g., V 9 CI(IR • It~",IR) is T-periodic in t and V ( t , O ) - 0, V q ( t , O ) - 0, then I 9 C t (E, R). A critical point of I on E is a classical solution of (HS2). Moreover q 9 E implies without loss of generality that q 9 C~IR, IR") and [q(t)l ~ 0 as it[ ~ cxz. Finally (HS2) implies ~ 9 L2(IR, IR") and therefore Iq(t)i --+ 0 as Itl --+ e~. Hence a critical point of I on E is a classical solution of (HS2) which is homoclinic to 0. Both minimizatir and minimax arguments have been employed to find homoclinics. An obstacle to both approaches is that I does not satisfy the (PS) condition. E.g., for q 9 E, set

rkq(t) = q(t - k).

(2.1.2)

I (rkq) = I (q)

(2.1.3)

Then

for all k e Z and q e E. Hence whenever (q,,,) is a (PS) sequence, so is (ro,,,q,,,) for all (0,,,) C Z. Thus a (PS) sequence will generally not have a convergent subsequence. For minimization arguments, one can get around this difficulty by normalizing the sequence, i.e., choosing 0,,, appropriately. However to find a nontrivial homoclinic, one needs to work with a class of functions that is not contractible and therefore the setup of the previous paragraph must be modified a bit.

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2.2. Some results f o r homoclinics As an example, consider a singular Hamiltonian system in ]~2. Results here can be obtained by elementary minimization arguments and serve as a paradigm for what is possible in more complex settings. Suppose V in (HS2) satisfies (Vg) V E C2(]t~ • ]I~2\{~:}, I~) and is T periodic in t with ~ # 0 .

(Vl0) l i m x ~ V ( t , x ) = - ~

uniformly for t E [0, T],

( V i i ) there is a neighborhood, A/" of ~ and U 6 CI (A/'\{~}, R) such that IU(x)l ~ as x --+ 0 and

o~

IU'(x)] 2 ~ < - v ( t , x ) for x E .Af\{~} and t E R, (Vl2) V ( t , 0 ) - - 0 > V ( t , x ) f o r x 6 1~2\{0}, Vq(t, O) -- O, and Vqq(t,O) is positive definite for all t E [0, T], (Vi3) there is a V0 < 0 and R > 0 such tllat V ( t , x ) <<,Vo for Ixl ~ R. Let

A = {q E E Iq(t) ~ ~ for all t E •}. Since q E A implies q(-t-~x~) - - 0 , q (R) is a closed curve which avoids ~. As such it has a winding number, W N (q), with respect t~ ~-. Let

F • -- {q E A I W N ( q ) E

-l-N}.

Now minimizing I in (2.1.1)over F + yields minimizers of I in F + and THEOREM 2.2.1 ([1 121). If V satisfies (Vg)-(VI3), (HS2) has a pair of solutions, homoclinic to O.

Q+,

REMARK 2.2.2. Here Q+ winds positively and Q - negatively around ~. When V is independent of t and satisfies another condition, Caldiroli and Nolasco [45] have shown that for any k E N, there is a solution, qk, of (HS2) which is homoclinic to 0 and with W N(qk) = k. Reversing time gives a solution with winding number -- - k . Caldiroli and Jeanjean [44] have also shown in the above setting that there is a solution of (HS2) which is heteroclinic to 0 and to a periodic solution of (HS2). This solution is obtained via a limiting process as k ----> oo from solutions qk with W N (qk) = k.

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P.H. Rabinowitz

Extensions of Theorem 2.2.1 have been made in other ways. Periodicity in t can be replaced by almost periodicity using indirect variational arguments introduced by Serra, Tarallo, and Terracini [121 ] in the setting of Theorem 2.2.4 below. Likewise asymptotic periodicity suffices. Moreover if, e.g., Q+ is an isolated minimizer, (HS2) can be perturbed by a suitably small term with arbitrary time dependence. As was mentioned earlier, the first variational results for homoclinic solutions of (HS) were carried out by Coti Zelati, Ekeland, and $6r6 [56] for superquadratic Hamiltonians. We will present an analogue of their results for the simpler setting of (HS2). Suppose V ( t , x ) = - L ( t ) x + W ( t , x ) where L and W satisfy: (L i) L 9 C(R, I~''2) is T-periodic and is a positive definite matrix for t e [0, T], (Wl) W 9 C 2(R x ]]~n, ]1~) and is T-periodic in t,

(w2) w(r,o)=o, wu(,,o)=o, wq~(t,o)=o, and (W3) there is a/z > 2 such that O < p W ( t , q ) <~q . W q ( t , q )

forq:ri0.

The associated Hamiltonian system is:

(2.2.3)

+ L(t)q + W q ( t , q ) - - 0

and one has: THEOREM 2.2.4. If L satisfies (Li) and W sati,sfies (WI)-(W3), then (HS2) has a nontrivial solution homoclinic to O. The functional corresponding to (2.2.3) is

I(q)-

f['~(I,~1-

+ L(t)q .q) - W ( t , q )

]

dt.

(2.2.5)

The assumptions on L and W imply that I is a functional of mountain pass type, i.e., 0 is a local but not global minimum of I. A proof of Theorem 2.2.4 was given in [ 109] finding the homoclinic solution as a limit of subharmonic solutions, i.e., as a limit of kT periodic solutions as k --+ o~. The periodic solutions were obtained via the Mountain Pass Theorem. A more direct approach can be found in [57]. When L = id, W is independent of t, and a further pinching condition on W is satisfied, Ambrosetti and Coti Zelati [8] found a second solution of (HS2) homoclinic to 0. See also Bessi [26] for a multiplicity result for homoclinics in another setting using comparison arguments.

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I l 13

As was the case with Theorem 2.2.2, there are many variants of Theorem 2.2.4. Under a further c o n d i t i o n - see ( . ) in Theorem 2.4.11 - among other things Alama and Li [ 1] showed that asymptotic periodicity in t suffices to get a homoclinic solution. As was mentioned earlier, Serra, Tarallo, and Terracini [ 121 ] weakened the periodicity requirement to almost periodicity. This was extended still further by Montecchiari, Nolasco, and Terracini [102] who simply require recurrence in t. Likewise under ( . ) of Theorem 2.4, Montecchiari and Nolasco [ 101 ] proved that the setting of Theorem 2.2.4 can be perturbed by a small term with arbitrary time dependence. Recently Ambrosetti and Badiale [5] have used a Lyapunov-Schmidt argument, i.e., a reduction to a finite dimensional problem which is then treated variationally to get existence results for homoclinics for some perturbation problems.

2.3. S o m e b a s i c h e t e r o c l i n i c r e s u l t s In this section a few results on basic heteroclinic solutions will be obtained. Suppose V satisfies (VI4)

V (E C2(]]{ >< ]~n, I~), is 7])-periodic in t, and

For convenience we take 7])= 1 : T/, 1 : i (Vis) 0 : V ( t , O )

Ti periodic in

x i , ] <~ i <~ n.

<~ n. Assume also

> V ( t , x ) , x 6 R " \ Z " and t 6 ]1{.

By (VI4)-(VIs), (HS2) has equilibrium solutions whenever x 6 Z". Thus one can seek heteroclinics joining a pair of these maxima of the potential V (or homoclinics to 0 if one views this as a problem on T"). Towards that end, let a, ~ 6 Z", a #_ ~, and F--{q

e

W 1"2 (]K, [{,t

I.c

)[q(-oo)--c~

,

q(o~)

:

/']1 .

By minimizing 1 in (2.1.1) over /-', it can be shown that an appropriately normalized minimizing sequence converges to a heteroclinic chain of solutions of (HS2). More precisely, THEOREM 2.3.1. I f V s a t i s f i e s (VI4)-(VI5), f o r a n y c~ :/: [4 ~ Z", t h e r e is a n i n t e g e r j -- j (lot-/31), h e t e r o c l i n i c s o l u t i o n s vi . . . . . v j o f (HS2)such t h a t vi ( - ~ ) = or, v i ( c x ) ) = v i + l ( -- o<) ) E Z " , 1 <. i <~ j -- 1, v i(cxz ) = [~ , a n d J

inf I -- ~ I"

Moreover

I (1)i). !

vi m i n i m i z e s

I over

{q 6 Wi1'2 I q ( - c x ) ) OC

vi(-cx))

~

q ( o o ) -- vi(oc))}

"

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This result was proved by Strobel [125]. Earlier versions are due to Kozlov [85] in the setting of the n-pendulum, in the autonomous case in an unpublished thesis of Bolotin (by less direct arguments) and in [ 107]. See also Felmer [68] and Cieliebak and $6r6 [48] for some related results for (HS). Next a more complicated example where the asymptotic states are both periodic solutions of (HS2) will be described. In fact results of this nature go back to the work of Morse [103] and Hedlund [77] on heteroclinic minimal geodesics. Suppose V satisfies ( g 14) and (Vl6) V is even in t (i.e., (HS2) is reversible). Then minimizing (2.3.2)

Ii (q) -- f0 T L ( q ) dt

1,2

over W T ([0, T]

, Rn ) produces a periodic solution v, of (HS2). Set

c = Ii (v)

(2.3.3)

1,2

and let .A4 denote the family of minimizers of Ii on W T , i.e., .A/[ = {q E W.).'2([0, T ] , • " ) l l l ( q ) - - c ] . Note that by (V 14), if q 6 .A4, so is q + k for any k 6 Z". Let A/" denote M modulo this Z" symmetry. Taking the simplest case, assume (./V') A/" consists of isolated points. (Note that (A/') fails if V is independent of t.) Then we have THEOREM 2.3.4 ([110]). S u p p o s e V sati,sfies (VI4), (VI6), a n d (A/') holds. Then f o r any v E .All, there is a w E A / l \ { v } a n d a solution, Q, o f (HS2) which is heteroclinic to v as t --+ - c ~ a n d to w as t ~

cx~.

The proof of Theorem 2.3.4 furnishes an example of the use of a renormalized functional. One cannot simply find Q by minimizing I in (2.1.1) over the class of curves, F , that are asymptotic to v as t ~ -cx~ and to some w 6 .A//\{v} as t ~ ~ since I will generally be infinite on such a curve. Instead, I must be replaced by a new functional J. For p 6 Z, define

/~

a p ( q ) --

f/~

L(q)dt -c,

-I

where c is as in (2.3.3) and set J (q) -- Z

pEZ

ap(q).

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Hypothesis (Vi6) implies ap(q) >/0 for all p ~ Z and q 6 F. Now minimizing J over F leads to Theorem 2.3.4. See also [ll0] where (N') is relaxed and see [32,35] for related results. Maxwell [100] has also shown that for any v 7~ w ~ .A4, there is a heteroclinic chain of solutions joining v and w. The periodic solutions of (HS2) in .A4 are even about T / 2 because of (V!6) and hence are contractible curves on T n . One can produce noncontractible periodic solutions of (HS2) on T". E.g., for a given homotopy type k 6 Z"\{0}, minimizing Ii over

{q ~ WI'2([O, TI, R '') I q ( t + T ) = q C t ) + k } gives a periodic solution of (HS2) of homotopy type k. It is not known whether there is an analogue of Theorem 2.3.4 for such classes of periodics. However some progress has been made in the autonomous case. Suppose

(V17) V E C2(/~n, I~) with

V l-periodic in xl . . . . . x,,

and (Vlg) V(0) = 0 > V ( x ) , x ~ R " \ Z " so Z" is a family of equilibrium solutions of (HS2). As above, one can try to find periodic solutions of homotopy type k by setting 1,2 , ~n )lthereisaT--T(q)suchthatq(t+T)--q(t)+k} F k - - { q 6 W h,c(IR

and minimizing

I~ (q) --

fo T(q) L(q)

dt

over Fk. Due to the presence of the equilibrium points, Z", there may not be a q E Fk minimizing Ii. E.g., a minimizing sequence might 'converge' to a heteroclinic joining 0 and k or to a heteroclinic chain. Hence one needs another hypothesis to go further. To formulate a geometrical such condition, set

Ck

infll l++k

and

f g'k-inf/L(q)dt, Gk

J~

where

Gk : {q E WIIo'2(N, ]R")I q ( - e c ) - - 0 ,

q ( o c ) - k}.

The elements of Gk are candidates for homoclinic solutions of (HS2). Results of Bolotin and Negrini [38] and of [ 114] show

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P.H. Rabinowitz

THEOREM 2.3.5. I f n -- 2, k = (kl,k2) with kl, k2 relatively prime, (g17)-(g18) hold, and ck < ~k,

(2.3.6)

then

(a) ck is achieved by q ~ Fk, a periodic solution o f (HS2), and (b) ck is achieved by u E Gk, a heteroclinic solution o f (HS2). Setting Pk-

{q ~ Fk ] Ii ( q ) - - c k ] ,

u lies between adjacent solutions v +, v - E Pk. Moreover setting

G

--

e G

lies between v + and

(u + ik) i=()

and

'4- - - i n f ~ L ( q ) d t ,

(c) there are integers [4+ >~ 2 such that f o r all [q ~ 1~, [4 >~ [4+ , c/~4 - k is achieved by 4-

+

q#k e G /~k, and

(d) there are solutions o f (HS2) heteroclini~" to 0 and v + (t), 0 and v - (t), 0 and v + ( - t ) , and 0 and v - ( - t ) . REMARK 2.3.7. All of the solutions of (a)-(d) have energy equal 0. They are also minimal solutions of (HS2) in the sense that if, e.g., v E Pk and cr < s, v minimizes

f

" L ( q ) dt

over the class of W 1'2 curves having q(cr) = v(o-) and q ( s ) = v(s). The proof of (d) in [38] involves geodesic and approximation arguments in the spirit of Morse [ 103] and Hedlund [77] while the proof in [ 114] uses a direct approach with a renormalized functional. For n > 2, analogues and extensions of Theorem 2.3.5 have been made recently under a further geometrical condition. See Bolotin and Rabinowitz [39]. More complicated variational constructions of connecting orbits have been carried out by Bolotin who found solutions homoclinic to invariant tori and to Mather sets [36,33].

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2.4. Multibump solutions: the time d e p e n d e n t case This section describes some of the progress made in using variational methods to find multibump solutions of (HS2). This area is evolving rapidly, both in terms of methods and results. Since the conclusions of the theorems one obtains here are closely related, only a small number of representative results will be stated and an indication of the ideas behind them will be given. To illustrate what is meant by a multibump solution, consider the settings of Theorem 2.2.1 or Theorem 2.2.4. As was noted earlier, since the potential V is l-periodic in t, whenever v is a homoclinic solution of (HS2), so is r.iv for any j 9 Z. Since any q 9 WI' 2 (R , ]]:~") is small near t --4-oo, e.g. , in II IIw~,, ,2U or II IIt~, if vl . . . . vk are homoclinic solutions of (HS2), then

k Qk

-- Z

(2.4.1)

r.jivi I

is a good approximate solution to the equation provided that the distance between the translates is sufficiently large. E.g., if jl < " " < jk, this is the case if ji+l - j i is large, 1 <~ i ~< k - 1. In the framework of Theorem 2.3.1, the analogue of (2.4.1) is an approximation to the heteroclinic chain joining given initial points c~,/4 9 Z". For the case of (2.4.1), a solution Q of (HS2) is called a k-bump homoclinic solution of the equation if Q is near ~-~ r i i vi. "Near" of course must be measured in an appropriate way. As an example, a precise statement will be given in the setting of Theorem 2.2. I. THEOREM 2.4.2 ([1 131). Assume V satisfies (V~))-(Vl3). Suppose that Q+ is an isolated critical point o f I in F +, i.e., ( . ) there is a p > 0 such that l ' ( q ) r

f o r a l l q with 0 < IIq - Q+ IIw~.-~/l~.ib~) < P.

Then there is an r() = r()(p) > 0 and g()(r) 9 I~ defined f o r 0 < r < r() such that whenever k 9 1~, and integers jl < "'" < jk satisfy .ji+l -- ji >~ g.o(r), 1 <~ i <~ k - 1, there exists a (kbump) homoclinic solution Q o f (HS2) and corresponding disjoint intervals Si . . . . . Sk C with $1, Sk semi-infinite and U~ si = R such that

II k

Q - ZrhQ I

< r,

+

1 ~< i ~< k.

(2.4.3)

W 1,2(Si, I~)

The intervals Si a r e chosen so that rji Q+ is (exponentially) small outside Si. The integer g0(r) is independent of k. Hence a straightforward limit process gives infinite bump solutions. Observe that if ( , ) fails, Q+ is a nonisolated homoclinic solution of (HS2). Thus if V satisfies (V9)-(VI3), (HS2) always has an infinite number of homoclinic solutions. If Q also satisfies (,), there is a more complicated set of solutions of (HS2) satisfying (2.4.3)

1 118

P.H. Rabinowitz

with Q+ replaced by Vi E {Q+, Q-}. There are also versions of Theorem 2.4.2 when the periodicity of V in t is weakened as in the remarks following Theorem 2.2.1. Theorem 2.4.2 and the remarks in the two paragraphs that follow it are typical of multibump results. Given hypotheses on V that guarantee a basic homoclinic or heteroclinic solution, and a condition like (,), then there are infinitely many multibump solutions. On the other hand, if ( , ) fails, by default, there are still an infinite number of homoclinics or heteroclinics. It is worth recalling the classical dynamical construction associated with homoclinics or heteroclinics. See, e.g., [84]. Thus suppose one has a periodically forced system of ordinary differential equations: x' = f (t, x)

(2.4.4)

with, e.g., 0 as an equilibrium solution and assume (2.4.4) has a solution s that is homoclinic to 0. Then for the associated Poincar6 map, ~ , 0 is a fixed point and there is a homoclinic point, ~, corresponding to s Suppose the stable and unstable manifolds for r through 0 intersect transversally at ,~. This requirement is the classical analogue of (,). In the terminology used above, the classical symbolic dynamics of solutions can be stated as: given any small r > 0 and any doubly infinite sequence ( j i ) < Z with j i + l -- ji > / g 0 ( r ) ,

(2.4.5)

there is a unique solution, x, of (2.4.4) such that

IIx - r i; ~ II 1,~ls, l ~ r, i.e., x shadows a prescribed set of translates of s This result is proved, e.g., via the Shadowing Lemma. For the approach presented here, variational arguments replace the Shadowing Lemma to get (2.4.3). Indeed some have called these arguments variational shadowing. The proof of the Shadowing Lemma plays back to the Contracting Mapping Theorem and leads to the uniqueness of x. This uniqueness property makes the map from the above class of doubly infinite sequences to the solution x a homeomorphism. It implies in particular that if ji+l - ji =-- ]

(2.4.6)

independently of i, then x is a periodic solution of (2.4.4). In contrast, for the variational construction when (2.4.6) is satisfied, corresponding s o l u t i o n s - there may not be uniqueness here - need not be periodic. However a separate construction can be made working in a class of periodic functions and resulting in a periodic solution. Conditions like transversality and ( , ) are difficult to verify. Sometimes the classical transversality requirement can be checked in perturbation problems using a Melnikov function. When transversality and ( , ) can be compared directly, ( , ) turns out to be a weaker condition.

Variational methods for Hamiltonian systems

1119

The extent to which ( , ) can be weakened has been studied in the setting of Theorem 2.2.1 in [ 115] and also by Cieliebak and $6r6 [49] in another setting. In particular in [ 115] it was shown that if KS denotes the set of global minimizers of I in F +, C the component of/C containing Q+, and S = {q(0) I q ~ C}, then either 0 6 S or a multibump construction can be carried out. Note that if ( , ) is autonomous, Q+ 6 C implies that roQ + 6 C for all 0 6 IR. Consequently Q+ ( - 0 ) ~ S for all 0 6 ]R so 0 6 S. Hence we get no information in the autonomous case. Some methods to treat autonomous problems will be described in Section 2.5. There are a small number of problems where multibump solutions have been obtained replacing (,) by other conditions such as compactness assumptions on the set of homoclinics, geometrical assumptions, or analyticity of the problem. See, e.g., Bolotin [33,34], and Buffoni and $6r6 [43]. We want to indicate some of the ideas behind the proof of Theorem 1.4.2. Let Br ( Q + ) denote an open ball of radius r about Q+ in E -- WI'2(R, IR2). By (,), there is a positive lower bound for III'11 in any annular region about Q+, say ]]l'(q)U ~>28(r)

forqEB4r(Q+)\Br/4(Q+).

Set Qk -= Y~= I rj; Q+ where the ji

are

(2.4.7)

widely spaced. Choose disjoint intervals Si so that

rj; Q+ is small outside of Si and U~ si - IR. Define a new equivalent norm on E via IIIqlll- max Ilqllw..2~s,.i~_~l and let/3,.(.) denote the corresponding ball of radius r in this norm. The inequality (2.4.7) leads to a lower bound for ][l'[I in an annular region about Qk (independently of the choice of jl < " " < jk satisfying (2.4.5)): [[l'(q)[[ ~>8

f o r q El~,.(Qk)\13,./2(Qk).

(2.4.8)

Moreover if the conclusion of Theorem 2.4.2 is false,

l'(q) ~ 0 ,

q E/3,-/2(Qk).

(2.4.9)

It remains to show that (2.4.8)-(2.4.9) lead to a contradiction. The part of the argument given thus far is common to multibump problems. Now the fact that Q+ is a (local) minimum of I will be used to find a local minimum of I near Qk. Consider/3,~ (Qk) where o" << r. Recall that I(Q +) =c. Hence l(q) is near kc for q e/3~ (Qk) via (2.4.1). If I has an interior local minimum in/3~, (Qk), the theorem is proved. Otherwise the infimum of I o v e r / 3 ~ ( Q k ) is attained at z e a/3~(Qk). Now a negative pseudogradient flow r/(s, z) for I can be constructed starting at z. The choice of z implies r/(s, z) cannot enter/3~(Qk) for s > 0. If for some ,~ > 0, r/(g, z) ~ a/3,.(Qk), (2.4.8) and the choice of cr imply (after some work!) that for some i, 1 <~ i ~< k,

f,s L(~/(,~, z)) dt < c - e i

(2.4.10)

P.H. Rabinowitz

1120

where s depends on r. But then r/(g, z)lsi can be modified to produce Q* ~ B r ( O +) with I ( Q * ) < c, contrary to the definition of c. Hence O ( s , z ) lies in 1 3 r ( Q k ) \ 1 3 a ( Q k ) for all s > 0. But then (2.4.8)-(2.4.9) show for s large, I (r/(s, z)) ~< 0, contrary to the form of I. In problems involving a minimax rather than a minimum, the latter part of the argument gets more complicated but the spirit is the same: a deformation is carried out producing a family of functions along which max I < c, with c the minimax value that one starts with. Next some other multibump results will be described. The setting that has received the most attention is the superquadratic case. This was studied first for (HS) by $6r6. Indeed [ 119] initiated the variational study of multibump solutions. However the simpler setting of Theorem 2.2.4 will be considered here for (HS2). Now let E = W 1,2 (~n, ~1;~)and let c denote the mountain pass minimax value associated with I, i.e., C

-

-

inf max I , gEF 0El0, I ]

- -,(g(O)

with I as in (2.2.5) and r = {g ~ c ( [ 0 , 1], E ) [ g ( 0 ) = 0

and l ( g ( l ) ) < 0}.

Let 1C' -- {q E E l l ( q )

~ s and l ' ( q ) -

0}.

Then it was shown in [571: THEOREM 2.4.1 1. S u p p o s e V - - - ~ LI ( t ) X .

X + W ( t , x ) sati,~fies (Li), ( W I ) - ( W 3 ) a n d

( , ) there is an ot > 0 such that IC'+~ is a finite set ( m o d u l o the Z" symmet~, o f I). Then: 1~ c is a critical value o f I, and 2 ~ fi~r any r > 0 a n d k ~ 1~, k ~ 2, a n d jl < "'" < jk ~ Z with .ji+l - j i > / g . o ( r ) , (2.2.3) has a m u l t i b u m p solution n e a r y ~ r.ii vi (in the sense o f Theorem 2 . 4 . 2 ) f o r any choice o f vi ~ lC - - {q E E I l (q) -- c a n d l ' ( q ) --O}.

The proof of Theorem 2.4.11 relies on minimax arguments and is much more complicated than that of Theorem 2.4.2. As with Theorem 2.4.2, there are also infinite bump solutions. Several authors have extended this result by weakening ( , ) and relaxing the time dependence of V on t (to asymptotically periodic, almost periodic, etc.). See, e.g., Alama and Li [1], Serra, Tarallo, and Terracini [121], Montecchiari and Nolasco [101], Montecchiari, Nolasco, and Terracini [ 102]. There is also a multibump result due to Strobel [125] in the setting of Theorem 2.3. Note that if vl . . . . . Vk is a heteroclinic chain joining ae and/4, so is Z'jl UI . . . . . r ik vk for any ji . . . . . jk E Z.

Variational methods for Hamiltonian systems

1121

THEOREM 2.4.12. Under the hypotheses of Theorem 2.3.1, suppose that: ( , ) vl . . . . . vk are each isolated solutions of (HS2).

Then there is an actual solution of (HS2) near the heteroclinic chain ~jl provided that ji + l -- ji is sufficiently large (independently of k).

UI . . . . .

"t'jk Uk

Again a limit argument gives infinite bump solutions. Note that there is a heteroclinic chain of solutions joining/3 to ct as well as the one joining ct and/~. Therefore if all of the "links" of the chain satisfy (,), there is a homoclinic chain joining c~ to ct. An extension of Theorem 2.4.12 to a class of related systems with almost periodic time dependence was carried out by Bertotti and Montecchiari [25]. See also Alessio, Bertotti, and Montecchiari [2]. There is also an analogue of Theorem 2.4.12 in the setting of Theorem 2.3.4 in an unpublished paper of Maxwell.

2.5. Multibumps in the autonomous case As was mentioned earlier, if (HS2) is autonomous, conditions like ( , ) in Theorems 2.4.2 and 2.4.11 fail. Therefore the multibump constructions indicated earlier cannot be carried out and indeed in the simplest cases phase plane arguments show there may only exist a small number of basic homoclinics or heteroclinics and no multibump solutions. However for some classes of problems further arguments have been found that yield multibump solutions. These new arguments generally combine variational methods with other tools, especially from the theory of dynamical systems. See, e.g., Bolotin [34], Buffoni and $6r6 [43], Berti and Bolle [24], and [39,40]. These papers deal with so-called saddlefocus or saddle-saddle situations. More precisely they study (HS2) or more general systems where 0 is a hyperbolic equilibrium point and also the characteristic exponents are either of the form -+-~l, -+-~.2 where ~.2 > )~l > 0 (the saddle-saddle case) or of the form -+-~ 4- i~t where ~,, ~ > 0 (the saddle-focus case). While the settings and approaches of [34,43,24,39,40] are different, their conclusions are all similar in spirit. Under an appropriate set of hypothesis, one gets a family of multibump homoclinic solutions as in Theorems 2.4.2 and 2.4.11. To illustrate, a saddle-focus problem is treated by Buffoni and $6r6 in [43]. The functional one encounters is of mountain pass type and it is not difficult to find a basic one bump homoclinic solution. The behavior of solutions of the variant of (HS2) studied here that are near the equilibrium point, 0, is known by dynamical systems arguments. A variational argument is then used to patch together solution pieces near and away from the origin to get k-bump homoclinic solutions. A limit procedure gives infinite bump solutions. Similar ideas occur in [40] where one has (V i 7)-(V 18) and (2.3.6) in addition to a saddlesaddle assumption. Then for small negative energies, there are k-bump periodic solutions of (HS2) near a concatenation of the basic heteroclinics of Theorem 2.3.5. The proof of this result uses a version of the ~.-lemma to control the solutions near {/3k I/3 ~ Z} and a minimization argument ties the solution pieces together. Again limit arguments give infinite bump solutions. Recently in the setting of Theorem 2.3.5, a complete symbolic dynamics

1122

PH. Rabinowitz

of 0 energy solutions of (HS2) has been obtained when (2.3.6) is strengthened [41 ]. The "finite-bump" solutions are asymptotic to v + or v - as t ---> - o ~ and as t ---> oo. However these solutions are not multibump solutions in that there is no analogue of, e.g., (2.4.3) for small r. In another recent paper, Berti and Bolle [24] also treat a saddle-saddle setting assuming the existence of two nondegenerate homoclinics together with additional assumptions. They use a mixture of a Lyapunov-Schmidt argument in the spirit of [5] and the ~,-lemma to reduce to a finite dimensional problem. This problem has a variational formulation and a degree theory argument is used to help solve it. One final piece of work that belongs in this section but involves somewhat different arguments is a paper of Kalies, Kwapisz, and VanderVorst [83]. They study a fourth order Hamiltonian system on R e. For a certain range of parameters, the system possesses a pair of saddle-focus equilibria. Direct minimization arguments combined with geometrical construction are employed to establish the existence of a rich set of homoclinic and heteroclinic orbits that make multiple transitions between and oscillations around the equilibrium points. Indeed one gets a broader set of orbits than multibump orbits. This is one of the few situations we know of where no analogue of ( , ) is required to get a complex symbolic dynamics of solutions.

3. Final remarks In this brief last section, a few open questions and future directions will be mentioned. For periodic solutions, the main open question remains that of understanding the multiplicity of such orbits on a given energy surface. E.g., in the setting of the EkelandLasry Theorem (Theorem 1.3.7), if one removes the pinching conditions (1.3.8), are there still at least n geometrically distinct solutions? Also virtually nothing is known about the multiplicity of periodics when H-Z(1) does not bound a convex region. Another area in which new ideas are needed is that of periodic solutions of singular Hamiltonian systems. The systems arising in celestial mechanics do not satisfy the strong force condition. Therefore the variational techniques as described in Section 1.9 at best give generalized solutions which may well be collision orbits. One would like more information about these solutions. Do they generally correspond to collisions? What is special about them? One area which remains almost open is a variational approach to almost periodic solutions of (HS) or (HS2). Virtually the only research we know of in the spirit of this survey consists of two papers of Berger and Chen [22,23]. However there has been a considerable amount of important work by John Mather from a different variational perspective. Mather's work deals with certain classes of quasiperiodic solutions and also with connecting orbits. See, e.g., [96-98]. It will be discussed in Chapter 13 of this volume. In the direction of homoclinic and heteroclinic orbits, it remains to find basic solutions in more complex settings, e.g., homoclinics or heteroclinics to invariant tori and to establish their multibump dynamics. Finally variational approaches to Arnold diffusion are just beginning to be explored. See, e.g., the recent work of Bessi [27-29]. There is also interesting work in progress in this direction by Mather.

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1123

References [1 ] S. Alama and Y.Y. Li, Existence of solutions for semilinear elliptic equations with indefinite linear part, J. Differential Equations 96 (1992), 88-115. [2] E Alessio, M.L. Bertotti and P. Montecchiari, Multibump solutions to possibly degenerate equilibria for almost periodic" Lagrangian systems, Preprint (1998). [3] A. Ambrosetti, Critical points and nonlinear variational problems, Mathematical Preprints of Sc. Norm. Sup., Vol. 118 (Nov. 1991 ). [4] A. Ambrosetti, Variational methods and nonlinear problems: classical results and recent advances, Topological Nonlinear Analysis, M. Matzeu and A. Vignoli, eds, Progress in Nonlinear Differential Equations and Their Applications, Vol. 15, Birkhfiuser (1995), 1-36. [5] A. Ambrosetti and M. Badiale, Homoclinics: Poincar~-Melnikov type results via a variational approach, Ann. Inst. H. Poincar6 Anal. Non Lin6aire, To appear. [6] A. Ambrosetti, V. Benci and Y. Long, A note on the existence of multiple brake orbits, Nonlinear Anal. 21 (1993), 643-649. [7] A. Ambrosetti and V. Coti Zelati, Critical points with lack of compactness and singular dynamical systems, Ann. Mat. Pura Appl. 149 (1987), 237-259. [8] A. Ambrosetti and V. Coti Zelati, Multiple homoclinic orbits fi)r a class of conservative systems, C. R. Acad. Sci. I-Math. 314 (1992), 601-604. [9] A. Ambrosetti and V. Coti Zelati, Periodic" Solutions of Singular Lagrangian Systems, Birkhfiuser (1993). [ 10] A. Ambrosetti and G. Mancini, Solutions of minimal period for a class of convex Hamiltonian systems, Math. Ann. 255 ( 1981 ), 405-42 i. [I i I H. Amann and E. Zehnder, Nontrivial solutions.for a class of nonresonance problems and applications to nonlinear d(ff'erential equations, Ann. Scuola Norm. Sup. Pisa CI. Sci. (4) 7 (1980), 539-603. !12] H. Amann and E. Zehnder, Periodic solutions ofasynlptotically linear Hamiltonian systems, Manuscripta Math. 32 (1980), 149-189. ! i 31 A. Bahri, Critical Points at h!finity in Some Variational Problems, Wiley, New York (1989). 1141 A. Bahri and H. Berestycki, Forced vibrations ofsuperquadratic Hamiltonian systems, Acta. Math. 152 (1984), 143-197. 1151 A. Bahri and H. Berestycki, ~:ristence of forced oscillations,fi~r some nonlinear differential equations, Comm. Pure Appl. Math. 37 (I 984), 403--442. 1161 A. Bahri and J.-M. Coron, On a nonlinear elliptic equation invoh,ing the critical Soholev exponent: the effect of the t~q~ology of the domain, Comm. Pure Appl. Math. 41 (1981), 253-294. [ 17 ] A. Bahri and P. Rabinowitz, A minimax method for a class of Hamiltonian systems with singular potentials, J. Funct. Anal. 82 (1989), 412-428. 1181 A. Bahri and P. Rabinowitz, Periodic solutions of Hamiltonian systems of 3-body type, Ann. Inst. H. Poincar6 Anal. Non Lin6aire 8 ( ! 991 ), 561-649. 1191 V. Benci, Closed geodesics.for the Jacobi metric and periodic solutions of prescribed energy oJ'natural Hamiltonian svsten~s, Ann. Inst. H. Poincar6 Anal. Non Lin6aire 1 (1984), 401-412. [201 V. Benci, H. Hofer and P.H. Rabinowitz, A remark on a priori bounds and existence for periodic solutions of Hantiltonian systems, Periodic Solutions of Hamiltonian Systems and Related Topics, P.H. Rabinowitz et al., eds, D. Reidel (1987), 85-88. 1211 H. Berestycki, J.-M. Lasry, G. Mancini and B. RuE Existence of multiple periodic" orbits on star-shaped Hamiltonian su~bc'es, Comm. Pure Appl. Math. 38 (1985), 253-290. 1221 M.S. Berger and Y.Y. Chen, Forced quasi-periodic and almost periodic, oscillations of nonlinear Duffing equations, Nonlinear Anal. 19 (1992), 249-257. [23] M.S. Berger and Y.Y. Chen, Forced quasi-periodic and ahnost-periodic solution ,fi~r nonlinear systems, Nonlinear Anal. 21 (1993), 949-965. [241 M. Berti and P. Bolle, Variational construction of homoclinics and chaos in presence of a saddle-saddle equilibrium, Preprint (I 998). I251 M.L. Bertotti and P. Montecchiari, Connecting orbits for some classes of almost periodic" Lagrangian systems, J. Differential Equations 145 (1998), 453-468. [26] U. Bessi, Multiple homoc'linics .fi)r autonomous singular potentials, Proc. Roy. Soc. Edinburgh A 124 (1994), 785-802.

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