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[792]1% This is LLNCS.DEM the demonstration file of
2% the LaTeX macro package from Springer-Verlag
3% for Lecture Notes in Computer Science, version 1.1
4\documentstyle{llncs}
5%
6\begin{document}
7
8\title{Hamiltonian Mechanics}
9
10\author{Ivar Ekeland\inst{1} and Roger Temam\inst{2}}
11
12\institute{Princeton University, Princeton NJ 08544, USA
13\and
14Universit\'{e} de Paris-Sud,
15Laboratoire d'Analyse Num\'{e}rique, B\^{a}timent 425,\\
16F-91405 Orsay Cedex, France}
17
18\maketitle
19
20\begin{abstract}
21The abstract should summarize the contents of the paper
22using at least 70 and at most 150 words. It will be set in 9-point
23font size and be inset 1.0 cm from the right and left margins.
24There will be two blank lines before and after the Abstract. \dots
25\end{abstract}
26%
27\section{Fixed-Period Problems: The Sublinear Case}
28%
29With this chapter, the preliminaries are over, and we begin the search
30for periodic solutions to Hamiltonian systems. All this will be done in
31the convex case; that is, we shall study the boundary-value problem
32\begin{eqnarray*}
33  \dot{x}&=&JH' (t,x)\\
34  x(0) &=& x(T)
35\end{eqnarray*}
36with $H(t,\cdot)$ a convex function of $x$, going to $+\infty$ when
37$\left\|x\right\| \to \infty$.
38
39%
40\subsection{Autonomous Systems}
41%
42In this section, we will consider the case when the Hamiltonian $H(x)$
43is autonomous. For the sake of simplicity, we shall also assume that it
44is $C^{1}$.
45
46We shall first consider the question of nontriviality, within the
47general framework of
48$\left(A_{\infty},B_{\infty}\right)$-subquadratic Hamiltonians. In
49the second subsection, we shall look into the special case when $H$ is
50$\left(0,b_{\infty}\right)$-subquadratic,
51and we shall try to derive additional information.
52%
53\subsubsection{ The General Case: Nontriviality.}
54%
55We assume that $H$ is
56$\left(A_{\infty},B_{\infty}\right)$-sub\-qua\-dra\-tic at infinity,
57for some constant symmetric matrices $A_{\infty}$ and $B_{\infty}$,
58with $B_{\infty}-A_{\infty}$ positive definite. Set:
59\begin{eqnarray}
60\gamma :&=&{\rm smallest\ eigenvalue\ of}\ \ B_{\infty} - A_{\infty} \\
61  \lambda : &=& {\rm largest\ negative\ eigenvalue\ of}\ \
62  J \frac{d}{dt} +A_{\infty}\ .
63\end{eqnarray}
64
65Theorem 21 tells us that if $\lambda +\gamma < 0$, the boundary-value
66problem:
67\begin{equation}
68\begin{array}{rcl}
69  \dot{x}&=&JH' (x)\\
70  x(0)&=&x (T)
71\end{array}
72\end{equation}
73has at least one solution
74$\overline{x}$, which is found by minimizing the dual
75action functional:
76\begin{equation}
77  \psi (u) = \int_{o}^{T} \left[\frac{1}{2}
78  \left(\Lambda_{o}^{-1} u,u\right) + N^{\ast} (-u)\right] dt
79\end{equation}
80on the range of $\Lambda$, which is a subspace $R (\Lambda)_{L}^{2}$
81with finite codimension. Here
82\begin{equation}
83  N(x) := H(x) - \frac{1}{2} \left(A_{\infty} x,x\right)
84\end{equation}
85is a convex function, and
86\begin{equation}
87  N(x) \le \frac{1}{2}
88  \left(\left(B_{\infty} - A_{\infty}\right) x,x\right)
89  + c\ \ \ \forall x\ .
90\end{equation}
91
92%
93\begin{proposition}
94Assume $H'(0)=0$ and $ H(0)=0$. Set:
95\begin{equation}
96  \delta := \liminf_{x\to 0} 2 N (x) \left\|x\right\|^{-2}\ .
97  \label{eq:one}
98\end{equation}
99
100If $\gamma < - \lambda < \delta$,
101the solution $\overline{u}$ is non-zero:
102\begin{equation}
103  \overline{x} (t) \ne 0\ \ \ \forall t\ .
104\end{equation}
105\end{proposition}
106%
107\begin{proof}
108Condition (\ref{eq:one}) means that, for every
109$\delta ' > \delta$, there is some $\varepsilon > 0$ such that
110\begin{equation}
111  \left\|x\right\| \le \varepsilon \Rightarrow N (x) \le
112  \frac{\delta '}{2} \left\|x\right\|^{2}\ .
113\end{equation}
114
115It is an exercise in convex analysis, into which we shall not go, to
116show that this implies that there is an $\eta > 0$ such that
117\begin{equation}
118  f\left\|x\right\| \le \eta
119  \Rightarrow N^{\ast} (y) \le \frac{1}{2\delta '}
120  \left\|y\right\|^{2}\ .
121  \label{eq:two}
122\end{equation}
123
124\begin{figure}
125\vspace{2.5cm}
126\caption{This is the caption of the figure displaying a white eagle and
127a white horse on a snow field}
128\end{figure}
129
130Since $u_{1}$ is a smooth function, we will have
131$\left\|hu_{1}\right\|_\infty \le \eta$
132for $h$ small enough, and inequality (\ref{eq:two}) will hold,
133yielding thereby:
134\begin{equation}
135  \psi (hu_{1}) \le \frac{h^{2}}{2}
136  \frac{1}{\lambda} \left\|u_{1} \right\|_{2}^{2} + \frac{h^{2}}{2}
137  \frac{1}{\delta '} \left\|u_{1}\right\|^{2}\ .
138\end{equation}
139
140If we choose $\delta '$ close enough to $\delta$, the quantity
141$\left(\frac{1}{\lambda} + \frac{1}{\delta '}\right)$
142will be negative, and we end up with
143\begin{equation}
144  \psi (hu_{1}) < 0\ \ \ \ \ {\rm for}\ \ h\ne 0\ \ {\rm small}\ .
145\end{equation}
146
147On the other hand, we check directly that $\psi (0) = 0$. This shows
148that 0 cannot be a minimizer of $\psi$, not even a local one.
149So $\overline{u} \ne 0$ and
150$\overline{u} \ne \Lambda_{o}^{-1} (0) = 0$. \qed
151\end{proof}
152%
153\begin{corollary}
154Assume $H$ is $C^{2}$ and
155$\left(a_{\infty},b_{\infty}\right)$-subquadratic at infinity. Let
156$\xi_{1},\allowbreak\dots,\allowbreak\xi_{N}$  be the
157equilibria, that is, the solutions of $H' (\xi ) = 0$.
158Denote by $\omega_{k}$
159the smallest eigenvalue of $H'' \left(\xi_{k}\right)$, and set:
160\begin{equation}
161  \omega : = {\rm Min\,} \left\{\omega_{1},\dots,\omega_{k}\right\}\ .
162\end{equation}
163If:
164\begin{equation}
165  \frac{T}{2\pi} b_{\infty} <
166  - E \left[- \frac{T}{2\pi}a_{\infty}\right] <
167  \frac{T}{2\pi}\omega
168  \label{eq:three}
169\end{equation}
170then minimization of $\psi$ yields a non-constant $T$-periodic solution
171$\overline{x}$.
172\end{corollary}
173%
174
175We recall once more that by the integer part $E [\alpha ]$ of
176$\alpha \in \bbbr$, we mean the $a\in \bbbz$
177such that $a< \alpha \le a+1$. For instance,
178if we take $a_{\infty} = 0$, Corollary 2 tells
179us that $\overline{x}$ exists and is
180non-constant provided that:
181
182\begin{equation}
183  \frac{T}{2\pi} b_{\infty} < 1 < \frac{T}{2\pi}
184\end{equation}
185or
186\begin{equation}
187  T\in \left(\frac{2\pi}{\omega},\frac{2\pi}{b_{\infty}}\right)\ .
188  \label{eq:four}
189\end{equation}
190
191%
192\begin{proof}
193The spectrum of $\Lambda$ is $\frac{2\pi}{T} \bbbz +a_{\infty}$. The
194largest negative eigenvalue $\lambda$ is given by
195$\frac{2\pi}{T}k_{o} +a_{\infty}$,
196where
197\begin{equation}
198  \frac{2\pi}{T}k_{o} + a_{\infty} < 0
199  \le \frac{2\pi}{T} (k_{o} +1) + a_{\infty}\ .
200\end{equation}
201Hence:
202\begin{equation}
203  k_{o} = E \left[- \frac{T}{2\pi} a_{\infty}\right] \ .
204\end{equation}
205
206The condition $\gamma < -\lambda < \delta$ now becomes:
207\begin{equation}
208  b_{\infty} - a_{\infty} <
209  - \frac{2\pi}{T} k_{o} -a_{\infty} < \omega -a_{\infty}
210\end{equation}
211which is precisely condition (\ref{eq:three}).\qed
212\end{proof}
213%
214
215\begin{lemma}
216Assume that $H$ is $C^{2}$ on $\bbbr^{2n} \setminus \{ 0\}$ and
217that $H'' (x)$ is non-de\-gen\-er\-ate for any $x\ne 0$. Then any local
218minimizer $\widetilde{x}$ of $\psi$ has minimal period $T$.
219\end{lemma}
220%
221\begin{proof}
222We know that $\widetilde{x}$, or
223$\widetilde{x} + \xi$ for some constant $\xi
224\in \bbbr^{2n}$, is a $T$-periodic solution of the Hamiltonian system:
225\begin{equation}
226  \dot{x} = JH' (x)\ .
227\end{equation}
228
229There is no loss of generality in taking $\xi = 0$. So
230$\psi (x) \ge \psi (\widetilde{x} )$
231for all $\widetilde{x}$ in some neighbourhood of $x$ in
232$W^{1,2} \left(\bbbr / T\bbbz ; \bbbr^{2n}\right)$.
233
234But this index is precisely the index
235$i_{T} (\widetilde{x} )$ of the $T$-periodic
236solution $\widetilde{x}$ over the interval
237$(0,T)$, as defined in Sect.~2.6. So
238\begin{equation}
239  i_{T} (\widetilde{x} ) = 0\ .
240  \label{eq:five}
241\end{equation}
242
243Now if $\widetilde{x}$ has a lower period, $T/k$ say,
244we would have, by Corollary 31:
245\begin{equation}
246  i_{T} (\widetilde{x} ) =
247  i_{kT/k}(\widetilde{x} ) \ge
248  ki_{T/k} (\widetilde{x} ) + k-1 \ge k-1 \ge 1\ .
249\end{equation}
250
251This would contradict (\ref{eq:five}), and thus cannot happen.\qed
252\end{proof}
253%
254\paragraph{Notes and Comments.}
255The results in this section are a
256refined version of \cite{clar:eke};
257the minimality result of Proposition
25814 was the first of its kind.
259
260To understand the nontriviality conditions, such as the one in formula
261(\ref{eq:four}), one may think of a one-parameter family
262$x_{T}$, $T\in \left(2\pi\omega^{-1}, 2\pi b_{\infty}^{-1}\right)$
263of periodic solutions, $x_{T} (0) = x_{T} (T)$,
264with $x_{T}$ going away to infinity when $T\to 2\pi \omega^{-1}$,
265which is the period of the linearized system at 0.
266
267\begin{table}
268\caption{This is the example table taken out of {\it The
269\TeX{}book,} p.\,246}
270\vspace{2pt}
271\begin{tabular}{r@{\quad}rl}
272\hline
273\multicolumn{1}{l}{\rule{0pt}{12pt}
274                   Year}&\multicolumn{2}{l}{World population}\\[2pt]
275\hline\rule{0pt}{12pt}
2768000 B.C.  &     5,000,000& \\
277  50 A.D.  &   200,000,000& \\
2781650 A.D.  &   500,000,000& \\
2791945 A.D.  & 2,300,000,000& \\
2801980 A.D.  & 4,400,000,000& \\[2pt]
281\hline
282\end{tabular}
283\end{table}
284%
285\begin{theorem} [(Ghoussoub-Preiss)]
286Assume $H(t,x)$ is
287$(0,\varepsilon )$-subquadratic at
288infinity for all $\varepsilon > 0$, and $T$-periodic in $t$
289\begin{equation}
290  H (t,\cdot )\ \ \ \ \ {\rm is\ convex}\ \ \forall t
291\end{equation}
292\begin{equation}
293  H (\cdot ,x)\ \ \ \ \ {\rm is}\ \ T{\rm -periodic}\ \ \forall x
294\end{equation}
295\begin{equation}
296  H (t,x)\ge n\left(\left\|x\right\|\right)\ \ \ \ \
297  {\rm with}\ \ n (s)s^{-1}\to \infty\ \ {\rm as}\ \ s\to \infty
298\end{equation}
299\begin{equation}
300  \forall \varepsilon > 0\ ,\ \ \ \exists c\ :\
301  H(t,x) \le \frac{\varepsilon}{2}\left\|x\right\|^{2} + c\ .
302\end{equation}
303
304Assume also that $H$ is $C^{2}$, and $H'' (t,x)$ is positive definite
305everywhere. Then there is a sequence $x_{k}$, $k\in \bbbn$, of
306$kT$-periodic solutions of the system
307\begin{equation}
308  \dot{x} = JH' (t,x)
309\end{equation}
310such that, for every $k\in \bbbn$, there is some $p_{o}\in\bbbn$ with:
311\begin{equation}
312  p\ge p_{o}\Rightarrow x_{pk} \ne x_{k}\ .
313\end{equation}
314\qed
315\end{theorem}
316%
317\begin{example} [{\rm(External forcing)}]
318Consider the system:
319\begin{equation}
320  \dot{x} = JH' (x) + f(t)
321\end{equation}
322where the Hamiltonian $H$ is
323$\left(0,b_{\infty}\right)$-subquadratic, and the
324forcing term is a distribution on the circle:
325\begin{equation}
326  f = \frac{d}{dt} F + f_{o}\ \ \ \ \
327  {\rm with}\ \ F\in L^{2} \left(\bbbr / T\bbbz; \bbbr^{2n}\right)\ ,
328\end{equation}
329where $f_{o} : = T^{-1}\int_{o}^{T} f (t) dt$. For instance,
330\begin{equation}
331  f (t) = \sum_{k\in \bbbn} \delta_{k} \xi\ ,
332\end{equation}
333where $\delta_{k}$ is the Dirac mass at $t= k$ and
334$\xi \in \bbbr^{2n}$ is a
335constant, fits the prescription. This means that the system
336$\dot{x} = JH' (x)$ is being excited by a
337series of identical shocks at interval $T$.
338\end{example}
339%
340\begin{definition}
341Let $A_{\infty} (t)$ and $B_{\infty} (t)$ be symmetric
342operators in $\bbbr^{2n}$, depending continuously on
343$t\in [0,T]$, such that
344$A_{\infty} (t) \le B_{\infty} (t)$ for all $t$.
345
346A Borelian function
347$H: [0,T]\times \bbbr^{2n} \to \bbbr$
348is called
349$\left(A_{\infty} ,B_{\infty}\right)$-{\it subquadratic at infinity}
350if there exists a function $N(t,x)$ such that:
351\begin{equation}
352  H (t,x) = \frac{1}{2} \left(A_{\infty} (t) x,x\right) + N(t,x)
353\end{equation}
354\begin{equation}
355  \forall t\ ,\ \ \ N(t,x)\ \ \ \ \
356  {\rm is\ convex\ with\  respect\  to}\ \ x
357\end{equation}
358\begin{equation}
359  N(t,x) \ge n\left(\left\|x\right\|\right)\ \ \ \ \
360  {\rm with}\ \ n(s)s^{-1}\to +\infty\ \ {\rm as}\ \ s\to +\infty
361\end{equation}
362\begin{equation}
363  \exists c\in \bbbr\ :\ \ \ H (t,x) \le
364  \frac{1}{2} \left(B_{\infty} (t) x,x\right) + c\ \ \ \forall x\ .
365\end{equation}
366
367If $A_{\infty} (t) = a_{\infty} I$ and
368$B_{\infty} (t) = b_{\infty} I$, with
369$a_{\infty} \le b_{\infty} \in \bbbr$,
370we shall say that $H$ is
371$\left(a_{\infty},b_{\infty}\right)$-subquadratic
372at infinity. As an example, the function
373$\left\|x\right\|^{\alpha}$, with
374$1\le \alpha < 2$, is $(0,\varepsilon )$-subquadratic at infinity
375for every $\varepsilon > 0$. Similarly, the Hamiltonian
376\begin{equation}
377H (t,x) = \frac{1}{2} k \left\|k\right\|^{2} +\left\|x\right\|^{\alpha}
378\end{equation}
379is $(k,k+\varepsilon )$-subquadratic for every $\varepsilon > 0$.
380Note that, if $k<0$, it is not convex.
381\end{definition}
382%
383
384\paragraph{Notes and Comments.}
385The first results on subharmonics were
386obtained by Rabinowitz in \cite{rab}, who showed the existence of
387infinitely many subharmonics both in the subquadratic and superquadratic
388case, with suitable growth conditions on $H'$. Again the duality
389approach enabled Clarke and Ekeland in \cite{clar:eke:2} to treat the
390same problem in the convex-subquadratic case, with growth conditions on
391$H$ only.
392
393Recently, Michalek and Tarantello (see \cite{mich:tar} and \cite{tar})
394have obtained lower bound on the number of subharmonics of period $kT$,
395based on symmetry considerations and on pinching estimates, as in
396Sect.~5.2 of this article.
397
398%
399% ---- Bibliography ----
400%
401\begin{thebibliography}{5}
402%
403\bibitem {clar:eke}
404Clarke, F., Ekeland, I.:
405Nonlinear oscillations and
406boundary-value problems for Hamiltonian systems.
407Arch. Rat. Mech. Anal. {\bf 78} (1982) 315--333
408%
409\bibitem {clar:eke:2}
410Clarke, F., Ekeland, I.:
411Solutions p\'{e}riodiques, du
412p\'{e}riode donn\'{e}e, des \'{e}quations hamiltoniennes.
413Note CRAS Paris {\bf 287} (1978) 1013--1015
414%
415\bibitem {mich:tar}
416Michalek, R., Tarantello, G.:
417Subharmonic solutions with prescribed minimal
418period for nonautonomous Hamiltonian systems.
419J. Diff. Eq. {\bf 72} (1988) 28--55
420%
421\bibitem {tar}
422Tarantello, G.:
423Subharmonic solutions for Hamiltonian
424systems via a $\bbbz_{p}$ pseudoindex theory.
425Annali di Matematica Pura (to appear)
426%
427\bibitem {rab}
428Rabinowitz, P.:
429On subharmonic solutions of a Hamiltonian system.
430Comm. Pure Appl. Math. {\bf 33} (1980) 609--633
431\end{thebibliography}
432%
433\end{document}
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