\contribution{Hamiltonian Mechanics \author{Ivar Ekeland@1 and Roger Temam@2 \address{@1Princeton University, Princeton, NJ 08544, USA @2Universit\’e de Paris-Sud, Laboratoire d’Analyse Num\’erique, B\^atiment 425, F-91405 Orsay Cedex, France \titlea{1{Fixed-Period Problems: The Sublinear Case With this chapter, the preliminaries are over, and we begin the search for periodic solutions to Hamiltonian systems. All this will be done in the convex case; that is, we shall study the boundary-value problem $$\eqalign{\dot x &= JH’ (t,x)\cr x(0) &= x(T)\cr$$ with $H(t,\bullet )$ a convex function of $x$, going to $+\un$ when $\lss x\rss \to \un$.

\titleb{1.1{Autonomous Systems In this section, we will consider the case when the Hamiltonian $H(x)$ is autonomous. For the sake of simplicity, we shall also assume that it is $C^1$.

We shall first consider the question of nontriviality, within the general framework of $\lr \Aun , \Bun\rr$-subquadratic Hamiltonians. In the second subsection, we shall look into the special case when $H$ is $\lr 0,\bun\rr$-subquadratic, and we shall try to derive additional information. \titlec{ The General Case: Nontriviality. We assume that $H$ is $\lr \Aun , \Bun \rr$-sub\-qua\-dra\-tic at infinity, for some constant symmetric matrices $\Aun$ and $\Bun$, with $\Bun -\Aun$ positive definite. Set: $$\eqalignno{ \gamma :& = {\rm smallest\ eigenvalue\ of\ \ \Bun - \Aun & (1)\cr \lambda : & = {\rm largest\ negative\ eigenvalue\ of\ \ J {d\ov dt +\Aun\ . & (2)\cr$$

Theorem 21 tells us that if $\lambda +\gamma < 0$, the boundary-value problem: $$\eqalign{ \dx &= JH’ (x)\cr x(0) &= x (T)\cr\eqno(3)$$ has at least one solution $\ol x$, which is found by minimizing the dual action functional: $$ \psi (u) = \int_o^T \lb \12 \lr \Lai_o^{-1 u,u\rr + N^\ast (-u)\rb dt\eqno(4)$$ on the range of $\Lai$, which is a subspace $R (\Lai )\sb L^2$ with finite codimension. Here $$ N(x) := H(x) - \12 \lr \Aun x,x\rr\eqno(5)$$ is a convex function, and $$ N(x) \le \12 \lr \lr \Bun - \Aun\rr x,x\rr + c\ \ \ \fa x\ .\eqno(6)$$

\proposition{ 1. { Assume $H’(0)=0$ and $ H(0)=0$. Set: $$ \de := \liminfu_{x\to 0 2 N (x) \lss x\rss^{-2\ .\eqno(7)$$

If $\gamma < - \lambda < \de$, the solution $\ol u$ is non-zero: $$ \ol x (t) \ne 0\ \ \ \fa t\ .\eqno(8)$$ \proof{ Condition (7) means that, for every $\de ’ > \de$, there is some $\ep > 0$ such that $$ \lss x\rss \le \ep \Rightarrow N (x) \le {\de ’\ov 2 \lss x\rss^2\ .\eqno(9)$$

It is an exercise in convex analysis, into which we shall not go, to show that this implies that there is an $\eta > 0$ such that $$ f\lss x\rss \le \eta \Rightarrow N^\ast (y) \le {1\ov 2\de ’ \lss y\rss^2\ .\eqno(10)$$

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Since $u_1$ is a smooth function, we will have $\lss hu_1\rss_\un \le \eta$ for $h$ small enough, and inequality (10) will hold, yielding thereby: $$ \psi (hu_1) \le {h^2\ov 2 {1\ov \lambda \lss u_1 \rss_2^2 + {h^2\ov 2 {1\ov \de ’ \lss u_1\rss^2\ .\eqno(11)$$

If we choose $\de ’$ close enough to $\de$, the quantity $\lr {1\ov \lambda + {1\ov \de ’\rr$ will be negative, and we end up with $$ \psi (hu_1) < 0\ \ \ \ \ \for\ \ h\ne 0\ \ {\rm small\ .\eqno(12)$$

On the other hand, we check directly that $\psi (0) = 0$. This shows that 0 cannot be a minimizer of $\psi$, not even a local one. So $\ol u \ne 0$ and $\ol u \ne \Lai_o^{-1 (0) = 0$. \qed

\corollary{ 2. { Assume $H$ is $C^2$ and $\lr \aun ,\bun\rr$-subquadratic at infinity. Let $\xi_1,\allowbreak\dots,\allowbreak\xi_N$ be the equilibria, that is, the solutions of $H’ (\xi ) = 0$. Denote by $\om_k$ the smallest eigenvalue of $H” \lr \xi_k\rr$, and set: $$ \om : = \Min \lg \om_1 , \dots , \om_k\rg\ .\eqno(13)$$ If: $$ {T\ov 2\pi \bun < - E \lb - {T\ov 2\pi\aun\rb < {T\ov 2\pi\om\eqno(14)$$ then minimization of $\psi$ yields a non-constant $T$-periodic solution $\ol x$. We recall once more that by the integer part $E [\al ]$ of $\al \in \RR$, we mean the $a\in \ZZ$ such that $a< \al \le a+1$. For instance, if we take $\aun = 0$, Corollary 2 tells us that $\ol x$ exists and is non-constant provided that: $$ {T\ov 2\pi \bun < 1 < {T\ov 2\pi\eqno(15)$$ or $$ T\in \lr {2\pi\ov \om,{2\pi\ov \bun\rr\ .\eqno(16)$$ \proof{ The spectrum of $\Lai$ is ${2\pi\ov T \ZZ +\aun$. The largest negative eigenvalue $\lambda$ is given by ${2\pi\ov Tk_o +\aun$, where $$ {2\pi\ov Tk_o + \aun < 0\le {2\pi\ov T (k_o +1) + \aun\ .\eqno(17)$$ Hence: $$ k_o = E \lb - {T\ov 2\pi \aun\rb \ .\eqno(18)$$

The condition $\gamma < -\lambda < \de$ now becomes: $$ \bun - \aun < - {2\pi\ov T k_o -\aun < \om -\aun\eqno(19)$$ which is precisely condition (14).\qed

\lemma {3. { Assume that $H$ is $C^2$ on $\RRn \sm \{ 0\$ and that $H” (x)$ is non-degenerate for any $x\ne 0$. Then any local minimizer $\tx$ of $\psi$ has minimal period $T$. \proof{ We know that $\tx$, or $\tx + \xi$ for some constant $\xi \in \RRn$, is a $T$-periodic solution of the Hamiltonian system: $$ \dx = JH’ (x)\ .\eqno(20)$$

There is no loss of generality in taking $\xi = 0$. So $\psi (x) \ge \psi (\tx )$ for all $\tx$ in some neighbourhood of $x$ in $W^{1,2 \lr \RR / T\ZZ ; \RRn\rr$.

But this index is precisely the index $i_T (\tx )$ of the $T$-periodic solution $\tx$ over the interval $(0,T)$, as defined in Sect.~2.6. So $$ i_T (\tx ) = 0\ .\eqno(21)$$

Now if $\tx$ has a lower period, $T/k$ say, we would have, by Corollary 31: $$ i_T (\tx ) = i_{kT/k(\tx ) \ge ki_{T/k (\tx ) + k-1 \ge k-1 \ge 1\ .\eqno(22)$$

This would contradict (21), and thus cannot happen.\qed \titled{Notes and Comments. The results in this section are a refined version of [CE1]; the minimality result of Proposition 14 was the first of its kind.

To understand the nontriviality conditions, such as the one in formula (16), one may think of a one-parameter family $x_T$, $T\in \lr 2\pi\om^{-1, 2\pi \bun^{-1\rr$ of periodic solutions, $x_T (0) = x_T (T)$, with $x_T$ going away to infinity when $T\to 2\pi \om^{-1$, which is the period of the linearized system at 0.

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\theorem{4 (Ghoussoub-Preiss). { Assume $H(t,x)$ is $(0,\ep )$-subquadratic at infinity for all $\ep > 0$, and $T$-periodic in $t$ $$ H (t,\bullet )\ \ \ \ \ {\rm is\ convex\ \ \fa t\eqno(23)$$ $$ H (\bullet ,x)\ \ \ \ \ {\rm is\ \ T{\rm -periodic\ \ \fa x \eqno(24)$$ $$ H (t,x)\ge n\lr \lss x\rss\rr\ \ \ \ \ {\rm with\ \ n (s)s^{-1\to \un\ \ {\rm as\ \ s\to \un\eqno(25)$$ $$ \fa \ep > 0\ ,\ \ \ \exists c\ :\ H(t,x) \le {\ep\ov 2\lss x\rss^2 + c\ .\eqno(26)$$

Assume also that $H$ is $C^2$, and $H” (t,x)$ is positive definite everywhere. Then there is a sequence $x_k$, $k\in \NN$, of $kT$-periodic solutions of the system $$ \dx = JH’ (t,x)\eqno(27)$$ such that, for every $k\in \NN$, there is some $p_o\in\NN$ with: $$ p\ge p_o\Rightarrow x_{pk \ne x_k\ .\eqno(28)$$\qed \example {1 {\rm(External forcing).{ Consider the system: $$ \dx = JH’ (x) + f(t)\eqno(29)$$ where the Hamiltonian $H$ is $\lr 0,\bun\rr$-subquadratic, and the forcing term is a distribution on the circle: $$ f = {d\ov dt F + f_o\ \ \ \ \ {\rm with\ \ F\in L^2 \lr \RR / T\ZZ ; \RRn\rr\ ,\eqno(30)$$ where $f_o : = T^{-1\int_o^T f (t) dt$. For instance, $$ f (t) = \sum_{k\in \NN \de_k \xi\ ,\eqno(31)$$ where $\de_k$ is the Dirac mass at $t= k$ and $\xi \in \RRn$ is a constant, fits the prescription. This means that the system $\dx = JH’ (x)$ is being excited by a series of identical shocks at interval $T$.

\definition{5.{Let $A_\un (t)$ and $B_\un (t)$ be symmetric operators in $\RRn$, depending continuously on $t\in [0,T]$, such that $A_\un (t) \le B_\un (t)$ for all $t$.

A Borelian function $H: [0,T]\times \RRn \to \RR$ is called $\lr A_\un ,B_\un\rr$-{\it subquadratic at infinity if there exists a function $N(t,x)$ such that: $$ H (t,x) = \12 \lr A_\un (t) x,x\rr + N(t,x)\eqno(32)$$ $$ \fa t\ ,\ \ \ N(t,x)\ \ \ \ \ {\rm is\ convex\ with\ respect\ to\ \ x\eqno(33)$$ $$ N(t,x) \ge n\lr \lss x\rss\rr\ \ \ \ \ {\rm with\ \ n(s)s^{-1\to +\un\ \ {\rm as\ \ s\to +\un\eqno(34)$$ $$ \exists c\in \RR\ :\ \ \ H (t,x) \le \12 \lr B_\un (t) x,x\rr + c\ \ \ \fa x\ .\eqno(35)$$ If $A_\un (t) = a_\un I$ and $B_\un (t) = b_\un I$, with $a_\un \le b_\un \in \RR$, we shall say that $H$ is $\lr a_\un ,b_\un\rr$-subquadratic at infinity. As an example, the function $\lss x \rss^\al$, with $1\le \al < 2$, is $(0,\ep )$-subquadratic at infinity for every $\ep > 0$. Similarly, the Hamiltonian $$ H (t,x) = \12 k \lss k\rss^2 +\lss x\rss^\al\eqno(36)$$ is $(k,k+\ep )$-subquadratic for every $\ep > 0$. Note that, if $k<0$, it is not convex.

\titled{Notes and Comments. The first results on subharmonics were obtained by Rabinowitz in [Ra1], 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 [CE2] to treat the same problem in the convex-subquadratic case, with growth conditions on $H$ only.

Recently, Michalek and Tarantello (see [MT1] and [Ta1]) 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 article. \begref{References{[MT1] \refmark{[CE1] Clarke, F., Ekeland, I.: Nonlinear oscillations and boundary-value problems for Hamiltonian systems. Arch. Rat. Mech. Anal. {\bf 78 (1982) 315–333 \refmark{[CE2] Clarke, F., Ekeland, I.: Solutions p\’eriodiques, du p\’eriode donn\’ee, des \’equations hamiltoiennes. Note CRAS Paris {\bf 287 (1978) 1013–1015 \refmark{[MT1] Michalek, R., Tarantello, G.: Subharmonic solutions with prescribed minimal period for nonautonomous Hamiltonian systems. J. Diff. Eq. {\bf 72 (1988) 28–55 \refmark{[Ta1] Tarantello, G.: Subharmonic solutions for Hamiltonian systems via a $\bbbz_p$ pseudoindex theory. Annali di Mathematica Pura (to appear) \refmark{[Ra1] Rabinowitz, P.: On subharmonic solutions of a Hamiltonian system. Comm. Pure Appl. Math. {\bf 33 (1980) 609–633 \endref \byebye

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