% This is LLNCS.DEM the demonstration file of % the LaTeX macro package from Springer-Verlag % for Lecture Notes in Computer Science, version 1.1 \documentstyle{llncs} % \begin{document} \title{Hamiltonian Mechanics} \author{Ivar Ekeland\inst{1} and Roger Temam\inst{2}} \institute{Princeton University, Princeton NJ 08544, USA \and Universit\'{e} de Paris-Sud, Laboratoire d'Analyse Num\'{e}rique, B\^{a}timent 425,\\ F-91405 Orsay Cedex, France} \maketitle \begin{abstract} The abstract should summarize the contents of the paper using at least 70 and at most 150 words. It will be set in 9-point font size and be inset 1.0 cm from the right and left margins. There will be two blank lines before and after the Abstract. \dots \end{abstract} % \section{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 \begin{eqnarray*} \dot{x}&=&JH' (t,x)\\ x(0) &=& x(T) \end{eqnarray*} with $H(t,\cdot)$ a convex function of $x$, going to $+\infty$ when $\left\|x\right\| \to \infty$. % \subsection{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 $\left(A_{\infty},B_{\infty}\right)$-subquadratic Hamiltonians. In the second subsection, we shall look into the special case when $H$ is $\left(0,b_{\infty}\right)$-subquadratic, and we shall try to derive additional information. % \subsubsection{ The General Case: Nontriviality.} % We assume that $H$ is $\left(A_{\infty},B_{\infty}\right)$-sub\-qua\-dra\-tic at infinity, for some constant symmetric matrices $A_{\infty}$ and $B_{\infty}$, with $B_{\infty}-A_{\infty}$ positive definite. Set: \begin{eqnarray} \gamma :&=&{\rm smallest\ eigenvalue\ of}\ \ B_{\infty} - A_{\infty} \\ \lambda : &=& {\rm largest\ negative\ eigenvalue\ of}\ \ J \frac{d}{dt} +A_{\infty}\ . \end{eqnarray} Theorem 21 tells us that if $\lambda +\gamma < 0$, the boundary-value problem: $$\begin{array}{rcl} \dot{x}&=&JH' (x)\\ x(0)&=&x (T) \end{array}$$ has at least one solution $\overline{x}$, which is found by minimizing the dual action functional: $$\psi (u) = \int_{o}^{T} \left[\frac{1}{2} \left(\Lambda_{o}^{-1} u,u\right) + N^{\ast} (-u)\right] dt$$ on the range of $\Lambda$, which is a subspace $R (\Lambda)_{L}^{2}$ with finite codimension. Here $$N(x) := H(x) - \frac{1}{2} \left(A_{\infty} x,x\right)$$ is a convex function, and $$N(x) \le \frac{1}{2} \left(\left(B_{\infty} - A_{\infty}\right) x,x\right) + c\ \ \ \forall x\ .$$ % \begin{proposition} Assume $H'(0)=0$ and $H(0)=0$. Set: $$\delta := \liminf_{x\to 0} 2 N (x) \left\|x\right\|^{-2}\ . \label{eq:one}$$ If $\gamma < - \lambda < \delta$, the solution $\overline{u}$ is non-zero: $$\overline{x} (t) \ne 0\ \ \ \forall t\ .$$ \end{proposition} % \begin{proof} Condition (\ref{eq:one}) means that, for every $\delta ' > \delta$, there is some $\varepsilon > 0$ such that $$\left\|x\right\| \le \varepsilon \Rightarrow N (x) \le \frac{\delta '}{2} \left\|x\right\|^{2}\ .$$ 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 $$f\left\|x\right\| \le \eta \Rightarrow N^{\ast} (y) \le \frac{1}{2\delta '} \left\|y\right\|^{2}\ . \label{eq:two}$$ \begin{figure} \vspace{2.5cm} \caption{This is the caption of the figure displaying a white eagle and a white horse on a snow field} \end{figure} Since $u_{1}$ is a smooth function, we will have $\left\|hu_{1}\right\|_\infty \le \eta$ for $h$ small enough, and inequality (\ref{eq:two}) will hold, yielding thereby: $$\psi (hu_{1}) \le \frac{h^{2}}{2} \frac{1}{\lambda} \left\|u_{1} \right\|_{2}^{2} + \frac{h^{2}}{2} \frac{1}{\delta '} \left\|u_{1}\right\|^{2}\ .$$ If we choose $\delta '$ close enough to $\delta$, the quantity $\left(\frac{1}{\lambda} + \frac{1}{\delta '}\right)$ will be negative, and we end up with $$\psi (hu_{1}) < 0\ \ \ \ \ {\rm for}\ \ h\ne 0\ \ {\rm small}\ .$$ 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 $\overline{u} \ne 0$ and $\overline{u} \ne \Lambda_{o}^{-1} (0) = 0$. \qed \end{proof} % \begin{corollary} Assume $H$ is $C^{2}$ and $\left(a_{\infty},b_{\infty}\right)$-subquadratic at infinity. Let $\xi_{1},\allowbreak\dots,\allowbreak\xi_{N}$ be the equilibria, that is, the solutions of $H' (\xi ) = 0$. Denote by $\omega_{k}$ the smallest eigenvalue of $H'' \left(\xi_{k}\right)$, and set: $$\omega : = {\rm Min\,} \left\{\omega_{1},\dots,\omega_{k}\right\}\ .$$ If: $$\frac{T}{2\pi} b_{\infty} < - E \left[- \frac{T}{2\pi}a_{\infty}\right] < \frac{T}{2\pi}\omega \label{eq:three}$$ then minimization of $\psi$ yields a non-constant $T$-periodic solution $\overline{x}$. \end{corollary} % We recall once more that by the integer part $E [\alpha ]$ of $\alpha \in \bbbr$, we mean the $a\in \bbbz$ such that $a< \alpha \le a+1$. For instance, if we take $a_{\infty} = 0$, Corollary 2 tells us that $\overline{x}$ exists and is non-constant provided that: $$\frac{T}{2\pi} b_{\infty} < 1 < \frac{T}{2\pi}$$ or $$T\in \left(\frac{2\pi}{\omega},\frac{2\pi}{b_{\infty}}\right)\ . \label{eq:four}$$ % \begin{proof} The spectrum of $\Lambda$ is $\frac{2\pi}{T} \bbbz +a_{\infty}$. The largest negative eigenvalue $\lambda$ is given by $\frac{2\pi}{T}k_{o} +a_{\infty}$, where $$\frac{2\pi}{T}k_{o} + a_{\infty} < 0 \le \frac{2\pi}{T} (k_{o} +1) + a_{\infty}\ .$$ Hence: $$k_{o} = E \left[- \frac{T}{2\pi} a_{\infty}\right] \ .$$ The condition $\gamma < -\lambda < \delta$ now becomes: $$b_{\infty} - a_{\infty} < - \frac{2\pi}{T} k_{o} -a_{\infty} < \omega -a_{\infty}$$ which is precisely condition (\ref{eq:three}).\qed \end{proof} % \begin{lemma} Assume that $H$ is $C^{2}$ on $\bbbr^{2n} \setminus \{ 0\}$ and that $H'' (x)$ is non-de\-gen\-er\-ate for any $x\ne 0$. Then any local minimizer $\widetilde{x}$ of $\psi$ has minimal period $T$. \end{lemma} % \begin{proof} We know that $\widetilde{x}$, or $\widetilde{x} + \xi$ for some constant $\xi \in \bbbr^{2n}$, is a $T$-periodic solution of the Hamiltonian system: $$\dot{x} = JH' (x)\ .$$ There is no loss of generality in taking $\xi = 0$. So $\psi (x) \ge \psi (\widetilde{x} )$ for all $\widetilde{x}$ in some neighbourhood of $x$ in $W^{1,2} \left(\bbbr / T\bbbz ; \bbbr^{2n}\right)$. But this index is precisely the index $i_{T} (\widetilde{x} )$ of the $T$-periodic solution $\widetilde{x}$ over the interval $(0,T)$, as defined in Sect.~2.6. So $$i_{T} (\widetilde{x} ) = 0\ . \label{eq:five}$$ Now if $\widetilde{x}$ has a lower period, $T/k$ say, we would have, by Corollary 31: $$i_{T} (\widetilde{x} ) = i_{kT/k}(\widetilde{x} ) \ge ki_{T/k} (\widetilde{x} ) + k-1 \ge k-1 \ge 1\ .$$ This would contradict (\ref{eq:five}), and thus cannot happen.\qed \end{proof} % \paragraph{Notes and Comments.} The results in this section are a refined version of \cite{clar:eke}; the minimality result of Proposition 14 was the first of its kind. To understand the nontriviality conditions, such as the one in formula (\ref{eq:four}), one may think of a one-parameter family $x_{T}$, $T\in \left(2\pi\omega^{-1}, 2\pi b_{\infty}^{-1}\right)$ of periodic solutions, $x_{T} (0) = x_{T} (T)$, with $x_{T}$ going away to infinity when $T\to 2\pi \omega^{-1}$, which is the period of the linearized system at 0. \begin{table} \caption{This is the example table taken out of {\it The \TeX{}book,} p.\,246} \vspace{2pt} \begin{tabular}{r@{\quad}rl} \hline \multicolumn{1}{l}{\rule{0pt}{12pt} Year}&\multicolumn{2}{l}{World population}\\[2pt] \hline\rule{0pt}{12pt} 8000 B.C. & 5,000,000& \\ 50 A.D. & 200,000,000& \\ 1650 A.D. & 500,000,000& \\ 1945 A.D. & 2,300,000,000& \\ 1980 A.D. & 4,400,000,000& \\[2pt] \hline \end{tabular} \end{table} % \begin{theorem} [(Ghoussoub-Preiss)] Assume $H(t,x)$ is $(0,\varepsilon )$-subquadratic at infinity for all $\varepsilon > 0$, and $T$-periodic in $t$ $$H (t,\cdot )\ \ \ \ \ {\rm is\ convex}\ \ \forall t$$ $$H (\cdot ,x)\ \ \ \ \ {\rm is}\ \ T{\rm -periodic}\ \ \forall x$$ $$H (t,x)\ge n\left(\left\|x\right\|\right)\ \ \ \ \ {\rm with}\ \ n (s)s^{-1}\to \infty\ \ {\rm as}\ \ s\to \infty$$ $$\forall \varepsilon > 0\ ,\ \ \ \exists c\ :\ H(t,x) \le \frac{\varepsilon}{2}\left\|x\right\|^{2} + c\ .$$ Assume also that $H$ is $C^{2}$, and $H'' (t,x)$ is positive definite everywhere. Then there is a sequence $x_{k}$, $k\in \bbbn$, of $kT$-periodic solutions of the system $$\dot{x} = JH' (t,x)$$ such that, for every $k\in \bbbn$, there is some $p_{o}\in\bbbn$ with: $$p\ge p_{o}\Rightarrow x_{pk} \ne x_{k}\ .$$ \qed \end{theorem} % \begin{example} [{\rm(External forcing)}] Consider the system: $$\dot{x} = JH' (x) + f(t)$$ where the Hamiltonian $H$ is $\left(0,b_{\infty}\right)$-subquadratic, and the forcing term is a distribution on the circle: $$f = \frac{d}{dt} F + f_{o}\ \ \ \ \ {\rm with}\ \ F\in L^{2} \left(\bbbr / T\bbbz; \bbbr^{2n}\right)\ ,$$ where $f_{o} : = T^{-1}\int_{o}^{T} f (t) dt$. For instance, $$f (t) = \sum_{k\in \bbbn} \delta_{k} \xi\ ,$$ where $\delta_{k}$ is the Dirac mass at $t= k$ and $\xi \in \bbbr^{2n}$ is a constant, fits the prescription. This means that the system $\dot{x} = JH' (x)$ is being excited by a series of identical shocks at interval $T$. \end{example} % \begin{definition} Let $A_{\infty} (t)$ and $B_{\infty} (t)$ be symmetric operators in $\bbbr^{2n}$, depending continuously on $t\in [0,T]$, such that $A_{\infty} (t) \le B_{\infty} (t)$ for all $t$. A Borelian function $H: [0,T]\times \bbbr^{2n} \to \bbbr$ is called $\left(A_{\infty} ,B_{\infty}\right)$-{\it subquadratic at infinity} if there exists a function $N(t,x)$ such that: $$H (t,x) = \frac{1}{2} \left(A_{\infty} (t) x,x\right) + N(t,x)$$ $$\forall t\ ,\ \ \ N(t,x)\ \ \ \ \ {\rm is\ convex\ with\ respect\ to}\ \ x$$ $$N(t,x) \ge n\left(\left\|x\right\|\right)\ \ \ \ \ {\rm with}\ \ n(s)s^{-1}\to +\infty\ \ {\rm as}\ \ s\to +\infty$$ $$\exists c\in \bbbr\ :\ \ \ H (t,x) \le \frac{1}{2} \left(B_{\infty} (t) x,x\right) + c\ \ \ \forall x\ .$$ If $A_{\infty} (t) = a_{\infty} I$ and $B_{\infty} (t) = b_{\infty} I$, with $a_{\infty} \le b_{\infty} \in \bbbr$, we shall say that $H$ is $\left(a_{\infty},b_{\infty}\right)$-subquadratic at infinity. As an example, the function $\left\|x\right\|^{\alpha}$, with $1\le \alpha < 2$, is $(0,\varepsilon )$-subquadratic at infinity for every $\varepsilon > 0$. Similarly, the Hamiltonian $$H (t,x) = \frac{1}{2} k \left\|k\right\|^{2} +\left\|x\right\|^{\alpha}$$ is $(k,k+\varepsilon )$-subquadratic for every $\varepsilon > 0$. Note that, if $k<0$, it is not convex. \end{definition} % \paragraph{Notes and Comments.} The first results on subharmonics were obtained by Rabinowitz in \cite{rab}, 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 \cite{clar:eke:2} to treat the same problem in the convex-subquadratic case, with growth conditions on $H$ only. Recently, Michalek and Tarantello (see \cite{mich:tar} and \cite{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 article. % % ---- Bibliography ---- % \begin{thebibliography}{5} % \bibitem {clar:eke} Clarke, F., Ekeland, I.: Nonlinear oscillations and boundary-value problems for Hamiltonian systems. Arch. Rat. Mech. Anal. {\bf 78} (1982) 315--333 % \bibitem {clar:eke:2} Clarke, F., Ekeland, I.: Solutions p\'{e}riodiques, du p\'{e}riode donn\'{e}e, des \'{e}quations hamiltoniennes. Note CRAS Paris {\bf 287} (1978) 1013--1015 % \bibitem {mich:tar} Michalek, R., Tarantello, G.: Subharmonic solutions with prescribed minimal period for nonautonomous Hamiltonian systems. J. Diff. Eq. {\bf 72} (1988) 28--55 % \bibitem {tar} Tarantello, G.: Subharmonic solutions for Hamiltonian systems via a $\bbbz_{p}$ pseudoindex theory. Annali di Matematica Pura (to appear) % \bibitem {rab} Rabinowitz, P.: On subharmonic solutions of a Hamiltonian system. Comm. Pure Appl. Math. {\bf 33} (1980) 609--633 \end{thebibliography} % \end{document}