initialisation

[prnggpu1.git] / llncs.dem

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% version 2.4 for LaTeX2e as of 16. April 2010\r
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\documentclass{llncs}\r
%\r
\usepackage{makeidx}  % allows for indexgeneration\r
%\r
\begin{document}\r
%\r
\frontmatter          % for the preliminaries\r
%\r
\pagestyle{headings}  % switches on printing of running heads\r
\addtocmark{Hamiltonian Mechanics} % additional mark in the TOC\r
%\r
\chapter*{Preface}\r
%\r
This textbook is intended for use by students of physics, physical\r
chemistry, and theoretical chemistry. The reader is presumed to have a\r
basic knowledge of atomic and quantum physics at the level provided, for\r
example, by the first few chapters in our book {\it The Physics of Atoms\r
and Quanta}. The student of physics will find here material which should\r
be included in the basic education of every physicist. This book should\r
furthermore allow students to acquire an appreciation of the breadth and\r
variety within the field of molecular physics and its future as a\r
fascinating area of research.\r
\r
For the student of chemistry, the concepts introduced in this book will\r
provide a theoretical framework for that entire field of study. With the\r
help of these concepts, it is at least in principle possible to reduce\r
the enormous body of empirical chemical knowledge to a few basic\r
principles: those of quantum mechanics. In addition, modern physical\r
methods whose fundamentals are introduced here are becoming increasingly\r
important in chemistry and now represent indispensable tools for the\r
chemist. As examples, we might mention the structural analysis of\r
complex organic compounds, spectroscopic investigation of very rapid\r
reaction processes or, as a practical application, the remote detection\r
of pollutants in the air.\r
\r
\vspace{1cm}\r
\begin{flushright}\noindent\r
April 1995\hfill Walter Olthoff\\\r
Program Chair\\\r
ECOOP'95\r
\end{flushright}\r
%\r
\chapter*{Organization}\r
ECOOP'95 is organized by the department of Computer Science, Univeristy\r
of \AA rhus and AITO (association Internationa pour les Technologie\r
Object) in cooperation with ACM/SIGPLAN.\r
%\r
\section*{Executive Commitee}\r
\begin{tabular}{@{}p{5cm}@{}p{7.2cm}@{}}\r
Conference Chair:&Ole Lehrmann Madsen (\AA rhus University, DK)\\\r
Program Chair:   &Walter Olthoff (DFKI GmbH, Germany)\\\r
Organizing Chair:&J\o rgen Lindskov Knudsen (\AA rhus University, DK)\\\r
Tutorials:&Birger M\o ller-Pedersen\hfil\break\r
(Norwegian Computing Center, Norway)\\\r
Workshops:&Eric Jul (University of Kopenhagen, Denmark)\\\r
Panels:&Boris Magnusson (Lund University, Sweden)\\\r
Exhibition:&Elmer Sandvad (\AA rhus University, DK)\\\r
Demonstrations:&Kurt N\o rdmark (\AA rhus University, DK)\r
\end{tabular}\r
%\r
\section*{Program Commitee}\r
\begin{tabular}{@{}p{5cm}@{}p{7.2cm}@{}}\r
Conference Chair:&Ole Lehrmann Madsen (\AA rhus University, DK)\\\r
Program Chair:   &Walter Olthoff (DFKI GmbH, Germany)\\\r
Organizing Chair:&J\o rgen Lindskov Knudsen (\AA rhus University, DK)\\\r
Tutorials:&Birger M\o ller-Pedersen\hfil\break\r
(Norwegian Computing Center, Norway)\\\r
Workshops:&Eric Jul (University of Kopenhagen, Denmark)\\\r
Panels:&Boris Magnusson (Lund University, Sweden)\\\r
Exhibition:&Elmer Sandvad (\AA rhus University, DK)\\\r
Demonstrations:&Kurt N\o rdmark (\AA rhus University, DK)\r
\end{tabular}\r
%\r
\begin{multicols}{3}[\section*{Referees}]\r
V.~Andreev\\\r
B\"arwolff\\\r
E.~Barrelet\\\r
H.P.~Beck\\\r
G.~Bernardi\\\r
E.~Binder\\\r
P.C.~Bosetti\\\r
Braunschweig\\\r
F.W.~B\"usser\\\r
T.~Carli\\\r
A.B.~Clegg\\\r
G.~Cozzika\\\r
S.~Dagoret\\\r
Del~Buono\\\r
P.~Dingus\\\r
H.~Duhm\\\r
J.~Ebert\\\r
S.~Eichenberger\\\r
R.J.~Ellison\\\r
Feltesse\\\r
W.~Flauger\\\r
A.~Fomenko\\\r
G.~Franke\\\r
J.~Garvey\\\r
M.~Gennis\\\r
L.~Goerlich\\\r
P.~Goritchev\\\r
H.~Greif\\\r
E.M.~Hanlon\\\r
R.~Haydar\\\r
R.C.W.~Henderso\\\r
P.~Hill\\\r
H.~Hufnagel\\\r
A.~Jacholkowska\\\r
Johannsen\\\r
S.~Kasarian\\\r
I.R.~Kenyon\\\r
C.~Kleinwort\\\r
T.~K\"ohler\\\r
S.D.~Kolya\\\r
P.~Kostka\\\r
U.~Kr\"uger\\\r
J.~Kurzh\"ofer\\\r
M.P.J.~Landon\\\r
A.~Lebedev\\\r
Ch.~Ley\\\r
F.~Linsel\\\r
H.~Lohmand\\\r
Martin\\\r
S.~Masson\\\r
K.~Meier\\\r
C.A.~Meyer\\\r
S.~Mikocki\\\r
J.V.~Morris\\\r
B.~Naroska\\\r
Nguyen\\\r
U.~Obrock\\\r
G.D.~Patel\\\r
Ch.~Pichler\\\r
S.~Prell\\\r
F.~Raupach\\\r
V.~Riech\\\r
P.~Robmann\\\r
N.~Sahlmann\\\r
P.~Schleper\\\r
Sch\"oning\\\r
B.~Schwab\\\r
A.~Semenov\\\r
G.~Siegmon\\\r
J.R.~Smith\\\r
M.~Steenbock\\\r
U.~Straumann\\\r
C.~Thiebaux\\\r
P.~Van~Esch\\\r
from Yerevan Ph\\\r
L.R.~West\\\r
G.-G.~Winter\\\r
T.P.~Yiou\\\r
M.~Zimmer\end{multicols}\r
%\r
\section*{Sponsoring Institutions}\r
%\r
Bernauer-Budiman Inc., Reading, Mass.\\\r
The Hofmann-International Company, San Louis Obispo, Cal.\\\r
Kramer Industries, Heidelberg, Germany\r
%\r
\tableofcontents\r
%\r
\mainmatter              % start of the contributions\r
%\r
\title{Hamiltonian Mechanics unter besonderer Ber\"ucksichtigung der\r
h\"ohreren Lehranstalten}\r
%\r
\titlerunning{Hamiltonian Mechanics}  % abbreviated title (for running head)\r
%                                     also used for the TOC unless\r
%                                     \toctitle is used\r
%\r
\author{Ivar Ekeland\inst{1} \and Roger Temam\inst{2}\r
Jeffrey Dean \and David Grove \and Craig Chambers \and Kim~B.~Bruce \and\r
Elsa Bertino}\r
%\r
\authorrunning{Ivar Ekeland et al.} % abbreviated author list (for running head)\r
%\r
%%%% list of authors for the TOC (use if author list has to be modified)\r
\tocauthor{Ivar Ekeland, Roger Temam, Jeffrey Dean, David Grove,\r
Craig Chambers, Kim B. Bruce, and Elisa Bertino}\r
%\r
\institute{Princeton University, Princeton NJ 08544, USA,\\\r
\email{I.Ekeland@princeton.edu},\\ WWW home page:\r
\texttt{http://users/\homedir iekeland/web/welcome.html}\r
\and\r
Universit\'{e} de Paris-Sud,\r
Laboratoire d'Analyse Num\'{e}rique, B\^{a}timent 425,\\\r
F-91405 Orsay Cedex, France}\r
\r
\maketitle              % typeset the title of the contribution\r
\r
\begin{abstract}\r
The abstract should summarize the contents of the paper\r
using at least 70 and at most 150 words. It will be set in 9-point\r
font size and be inset 1.0 cm from the right and left margins.\r
There will be two blank lines before and after the Abstract. \dots\r
\keywords{computational geometry, graph theory, Hamilton cycles}\r
\end{abstract}\r
%\r
\section{Fixed-Period Problems: The Sublinear Case}\r
%\r
With this chapter, the preliminaries are over, and we begin the search\r
for periodic solutions to Hamiltonian systems. All this will be done in\r
the convex case; that is, we shall study the boundary-value problem\r
\begin{eqnarray*}\r
  \dot{x}&=&JH' (t,x)\\\r
  x(0) &=& x(T)\r
\end{eqnarray*}\r
with $H(t,\cdot)$ a convex function of $x$, going to $+\infty$ when\r
$\left\|x\right\| \to \infty$.\r
\r
%\r
\subsection{Autonomous Systems}\r
%\r
In this section, we will consider the case when the Hamiltonian $H(x)$\r
is autonomous. For the sake of simplicity, we shall also assume that it\r
is $C^{1}$.\r
\r
We shall first consider the question of nontriviality, within the\r
general framework of\r
$\left(A_{\infty},B_{\infty}\right)$-subquadratic Hamiltonians. In\r
the second subsection, we shall look into the special case when $H$ is\r
$\left(0,b_{\infty}\right)$-subquadratic,\r
and we shall try to derive additional information.\r
%\r
\subsubsection{The General Case: Nontriviality.}\r
%\r
We assume that $H$ is\r
$\left(A_{\infty},B_{\infty}\right)$-sub\-qua\-dra\-tic at infinity,\r
for some constant symmetric matrices $A_{\infty}$ and $B_{\infty}$,\r
with $B_{\infty}-A_{\infty}$ positive definite. Set:\r
\begin{eqnarray}\r
\gamma :&=&{\rm smallest\ eigenvalue\ of}\ \ B_{\infty} - A_{\infty} \\\r
  \lambda : &=& {\rm largest\ negative\ eigenvalue\ of}\ \\r
  J \frac{d}{dt} +A_{\infty}\ .\r
\end{eqnarray}\r
\r
Theorem~\ref{ghou:pre} tells us that if $\lambda +\gamma < 0$, the\r
boundary-value problem:\r
\begin{equation}\r
\begin{array}{rcl}\r
  \dot{x}&=&JH' (x)\\\r
  x(0)&=&x (T)\r
\end{array}\r
\end{equation}\r
has at least one solution\r
$\overline{x}$, which is found by minimizing the dual\r
action functional:\r
\begin{equation}\r
  \psi (u) = \int_{o}^{T} \left[\frac{1}{2}\r
  \left(\Lambda_{o}^{-1} u,u\right) + N^{\ast} (-u)\right] dt\r
\end{equation}\r
on the range of $\Lambda$, which is a subspace $R (\Lambda)_{L}^{2}$\r
with finite codimension. Here\r
\begin{equation}\r
  N(x) := H(x) - \frac{1}{2} \left(A_{\infty} x,x\right)\r
\end{equation}\r
is a convex function, and\r
\begin{equation}\r
  N(x) \le \frac{1}{2}\r
  \left(\left(B_{\infty} - A_{\infty}\right) x,x\right)\r
  + c\ \ \ \forall x\ .\r
\end{equation}\r
\r
%\r
\begin{proposition}\r
Assume $H'(0)=0$ and $ H(0)=0$. Set:\r
\begin{equation}\r
  \delta := \liminf_{x\to 0} 2 N (x) \left\|x\right\|^{-2}\ .\r
  \label{eq:one}\r
\end{equation}\r
\r
If $\gamma < - \lambda < \delta$,\r
the solution $\overline{u}$ is non-zero:\r
\begin{equation}\r
  \overline{x} (t) \ne 0\ \ \ \forall t\ .\r
\end{equation}\r
\end{proposition}\r
%\r
\begin{proof}\r
Condition (\ref{eq:one}) means that, for every\r
$\delta ' > \delta$, there is some $\varepsilon > 0$ such that\r
\begin{equation}\r
  \left\|x\right\| \le \varepsilon \Rightarrow N (x) \le\r
  \frac{\delta '}{2} \left\|x\right\|^{2}\ .\r
\end{equation}\r
\r
It is an exercise in convex analysis, into which we shall not go, to\r
show that this implies that there is an $\eta > 0$ such that\r
\begin{equation}\r
  f\left\|x\right\| \le \eta\r
  \Rightarrow N^{\ast} (y) \le \frac{1}{2\delta '}\r
  \left\|y\right\|^{2}\ .\r
  \label{eq:two}\r
\end{equation}\r
\r
\begin{figure}\r
\vspace{2.5cm}\r
\caption{This is the caption of the figure displaying a white eagle and\r
a white horse on a snow field}\r
\end{figure}\r
\r
Since $u_{1}$ is a smooth function, we will have\r
$\left\|hu_{1}\right\|_\infty \le \eta$\r
for $h$ small enough, and inequality (\ref{eq:two}) will hold,\r
yielding thereby:\r
\begin{equation}\r
  \psi (hu_{1}) \le \frac{h^{2}}{2}\r
  \frac{1}{\lambda} \left\|u_{1} \right\|_{2}^{2} + \frac{h^{2}}{2}\r
  \frac{1}{\delta '} \left\|u_{1}\right\|^{2}\ .\r
\end{equation}\r
\r
If we choose $\delta '$ close enough to $\delta$, the quantity\r
$\left(\frac{1}{\lambda} + \frac{1}{\delta '}\right)$\r
will be negative, and we end up with\r
\begin{equation}\r
  \psi (hu_{1}) < 0\ \ \ \ \ {\rm for}\ \ h\ne 0\ \ {\rm small}\ .\r
\end{equation}\r
\r
On the other hand, we check directly that $\psi (0) = 0$. This shows\r
that 0 cannot be a minimizer of $\psi$, not even a local one.\r
So $\overline{u} \ne 0$ and\r
$\overline{u} \ne \Lambda_{o}^{-1} (0) = 0$. \qed\r
\end{proof}\r
%\r
\begin{corollary}\r
Assume $H$ is $C^{2}$ and\r
$\left(a_{\infty},b_{\infty}\right)$-subquadratic at infinity. Let\r
$\xi_{1},\allowbreak\dots,\allowbreak\xi_{N}$  be the\r
equilibria, that is, the solutions of $H' (\xi ) = 0$.\r
Denote by $\omega_{k}$\r
the smallest eigenvalue of $H'' \left(\xi_{k}\right)$, and set:\r
\begin{equation}\r
  \omega : = {\rm Min\,} \left\{\omega_{1},\dots,\omega_{k}\right\}\ .\r
\end{equation}\r
If:\r
\begin{equation}\r
  \frac{T}{2\pi} b_{\infty} <\r
  - E \left[- \frac{T}{2\pi}a_{\infty}\right] <\r
  \frac{T}{2\pi}\omega\r
  \label{eq:three}\r
\end{equation}\r
then minimization of $\psi$ yields a non-constant $T$-periodic solution\r
$\overline{x}$.\r
\end{corollary}\r
%\r
\r
We recall once more that by the integer part $E [\alpha ]$ of\r
$\alpha \in \bbbr$, we mean the $a\in \bbbz$\r
such that $a< \alpha \le a+1$. For instance,\r
if we take $a_{\infty} = 0$, Corollary 2 tells\r
us that $\overline{x}$ exists and is\r
non-constant provided that:\r
\r
\begin{equation}\r
  \frac{T}{2\pi} b_{\infty} < 1 < \frac{T}{2\pi}\r
\end{equation}\r
or\r
\begin{equation}\r
  T\in \left(\frac{2\pi}{\omega},\frac{2\pi}{b_{\infty}}\right)\ .\r
  \label{eq:four}\r
\end{equation}\r
\r
%\r
\begin{proof}\r
The spectrum of $\Lambda$ is $\frac{2\pi}{T} \bbbz +a_{\infty}$. The\r
largest negative eigenvalue $\lambda$ is given by\r
$\frac{2\pi}{T}k_{o} +a_{\infty}$,\r
where\r
\begin{equation}\r
  \frac{2\pi}{T}k_{o} + a_{\infty} < 0\r
  \le \frac{2\pi}{T} (k_{o} +1) + a_{\infty}\ .\r
\end{equation}\r
Hence:\r
\begin{equation}\r
  k_{o} = E \left[- \frac{T}{2\pi} a_{\infty}\right] \ .\r
\end{equation}\r
\r
The condition $\gamma < -\lambda < \delta$ now becomes:\r
\begin{equation}\r
  b_{\infty} - a_{\infty} <\r
  - \frac{2\pi}{T} k_{o} -a_{\infty} < \omega -a_{\infty}\r
\end{equation}\r
which is precisely condition (\ref{eq:three}).\qed\r
\end{proof}\r
%\r
\r
\begin{lemma}\r
Assume that $H$ is $C^{2}$ on $\bbbr^{2n} \setminus \{ 0\}$ and\r
that $H'' (x)$ is non-de\-gen\-er\-ate for any $x\ne 0$. Then any local\r
minimizer $\widetilde{x}$ of $\psi$ has minimal period $T$.\r
\end{lemma}\r
%\r
\begin{proof}\r
We know that $\widetilde{x}$, or\r
$\widetilde{x} + \xi$ for some constant $\xi\r
\in \bbbr^{2n}$, is a $T$-periodic solution of the Hamiltonian system:\r
\begin{equation}\r
  \dot{x} = JH' (x)\ .\r
\end{equation}\r
\r
There is no loss of generality in taking $\xi = 0$. So\r
$\psi (x) \ge \psi (\widetilde{x} )$\r
for all $\widetilde{x}$ in some neighbourhood of $x$ in\r
$W^{1,2} \left(\bbbr / T\bbbz ; \bbbr^{2n}\right)$.\r
\r
But this index is precisely the index\r
$i_{T} (\widetilde{x} )$ of the $T$-periodic\r
solution $\widetilde{x}$ over the interval\r
$(0,T)$, as defined in Sect.~2.6. So\r
\begin{equation}\r
  i_{T} (\widetilde{x} ) = 0\ .\r
  \label{eq:five}\r
\end{equation}\r
\r
Now if $\widetilde{x}$ has a lower period, $T/k$ say,\r
we would have, by Corollary 31:\r
\begin{equation}\r
  i_{T} (\widetilde{x} ) =\r
  i_{kT/k}(\widetilde{x} ) \ge\r
  ki_{T/k} (\widetilde{x} ) + k-1 \ge k-1 \ge 1\ .\r
\end{equation}\r
\r
This would contradict (\ref{eq:five}), and thus cannot happen.\qed\r
\end{proof}\r
%\r
\paragraph{Notes and Comments.}\r
The results in this section are a\r
refined version of \cite{clar:eke};\r
the minimality result of Proposition\r
14 was the first of its kind.\r
\r
To understand the nontriviality conditions, such as the one in formula\r
(\ref{eq:four}), one may think of a one-parameter family\r
$x_{T}$, $T\in \left(2\pi\omega^{-1}, 2\pi b_{\infty}^{-1}\right)$\r
of periodic solutions, $x_{T} (0) = x_{T} (T)$,\r
with $x_{T}$ going away to infinity when $T\to 2\pi \omega^{-1}$,\r
which is the period of the linearized system at 0.\r
\r
\begin{table}\r
\caption{This is the example table taken out of {\it The\r
\TeX{}book,} p.\,246}\r
\begin{center}\r
\begin{tabular}{r@{\quad}rl}\r
\hline\r
\multicolumn{1}{l}{\rule{0pt}{12pt}\r
                   Year}&\multicolumn{2}{l}{World population}\\[2pt]\r
\hline\rule{0pt}{12pt}\r
8000 B.C.  &     5,000,000& \\\r
  50 A.D.  &   200,000,000& \\\r
1650 A.D.  &   500,000,000& \\\r
1945 A.D.  & 2,300,000,000& \\\r
1980 A.D.  & 4,400,000,000& \\[2pt]\r
\hline\r
\end{tabular}\r
\end{center}\r
\end{table}\r
%\r
\begin{theorem} [Ghoussoub-Preiss]\label{ghou:pre}\r
Assume $H(t,x)$ is\r
$(0,\varepsilon )$-subquadratic at\r
infinity for all $\varepsilon > 0$, and $T$-periodic in $t$\r
\begin{equation}\r
  H (t,\cdot )\ \ \ \ \ {\rm is\ convex}\ \ \forall t\r
\end{equation}\r
\begin{equation}\r
  H (\cdot ,x)\ \ \ \ \ {\rm is}\ \ T{\rm -periodic}\ \ \forall x\r
\end{equation}\r
\begin{equation}\r
  H (t,x)\ge n\left(\left\|x\right\|\right)\ \ \ \ \\r
  {\rm with}\ \ n (s)s^{-1}\to \infty\ \ {\rm as}\ \ s\to \infty\r
\end{equation}\r
\begin{equation}\r
  \forall \varepsilon > 0\ ,\ \ \ \exists c\ :\\r
  H(t,x) \le \frac{\varepsilon}{2}\left\|x\right\|^{2} + c\ .\r
\end{equation}\r
\r
Assume also that $H$ is $C^{2}$, and $H'' (t,x)$ is positive definite\r
everywhere. Then there is a sequence $x_{k}$, $k\in \bbbn$, of\r
$kT$-periodic solutions of the system\r
\begin{equation}\r
  \dot{x} = JH' (t,x)\r
\end{equation}\r
such that, for every $k\in \bbbn$, there is some $p_{o}\in\bbbn$ with:\r
\begin{equation}\r
  p\ge p_{o}\Rightarrow x_{pk} \ne x_{k}\ .\r
\end{equation}\r
\qed\r
\end{theorem}\r
%\r
\begin{example} [{{\rm External forcing}}]\r
Consider the system:\r
\begin{equation}\r
  \dot{x} = JH' (x) + f(t)\r
\end{equation}\r
where the Hamiltonian $H$ is\r
$\left(0,b_{\infty}\right)$-subquadratic, and the\r
forcing term is a distribution on the circle:\r
\begin{equation}\r
  f = \frac{d}{dt} F + f_{o}\ \ \ \ \\r
  {\rm with}\ \ F\in L^{2} \left(\bbbr / T\bbbz; \bbbr^{2n}\right)\ ,\r
\end{equation}\r
where $f_{o} : = T^{-1}\int_{o}^{T} f (t) dt$. For instance,\r
\begin{equation}\r
  f (t) = \sum_{k\in \bbbn} \delta_{k} \xi\ ,\r
\end{equation}\r
where $\delta_{k}$ is the Dirac mass at $t= k$ and\r
$\xi \in \bbbr^{2n}$ is a\r
constant, fits the prescription. This means that the system\r
$\dot{x} = JH' (x)$ is being excited by a\r
series of identical shocks at interval $T$.\r
\end{example}\r
%\r
\begin{definition}\r
Let $A_{\infty} (t)$ and $B_{\infty} (t)$ be symmetric\r
operators in $\bbbr^{2n}$, depending continuously on\r
$t\in [0,T]$, such that\r
$A_{\infty} (t) \le B_{\infty} (t)$ for all $t$.\r
\r
A Borelian function\r
$H: [0,T]\times \bbbr^{2n} \to \bbbr$\r
is called\r
$\left(A_{\infty} ,B_{\infty}\right)$-{\it subquadratic at infinity}\r
if there exists a function $N(t,x)$ such that:\r
\begin{equation}\r
  H (t,x) = \frac{1}{2} \left(A_{\infty} (t) x,x\right) + N(t,x)\r
\end{equation}\r
\begin{equation}\r
  \forall t\ ,\ \ \ N(t,x)\ \ \ \ \\r
  {\rm is\ convex\ with\  respect\  to}\ \ x\r
\end{equation}\r
\begin{equation}\r
  N(t,x) \ge n\left(\left\|x\right\|\right)\ \ \ \ \\r
  {\rm with}\ \ n(s)s^{-1}\to +\infty\ \ {\rm as}\ \ s\to +\infty\r
\end{equation}\r
\begin{equation}\r
  \exists c\in \bbbr\ :\ \ \ H (t,x) \le\r
  \frac{1}{2} \left(B_{\infty} (t) x,x\right) + c\ \ \ \forall x\ .\r
\end{equation}\r
\r
If $A_{\infty} (t) = a_{\infty} I$ and\r
$B_{\infty} (t) = b_{\infty} I$, with\r
$a_{\infty} \le b_{\infty} \in \bbbr$,\r
we shall say that $H$ is\r
$\left(a_{\infty},b_{\infty}\right)$-subquadratic\r
at infinity. As an example, the function\r
$\left\|x\right\|^{\alpha}$, with\r
$1\le \alpha < 2$, is $(0,\varepsilon )$-subquadratic at infinity\r
for every $\varepsilon > 0$. Similarly, the Hamiltonian\r
\begin{equation}\r
H (t,x) = \frac{1}{2} k \left\|k\right\|^{2} +\left\|x\right\|^{\alpha}\r
\end{equation}\r
is $(k,k+\varepsilon )$-subquadratic for every $\varepsilon > 0$.\r
Note that, if $k<0$, it is not convex.\r
\end{definition}\r
%\r
\r
\paragraph{Notes and Comments.}\r
The first results on subharmonics were\r
obtained by Rabinowitz in \cite{rab}, who showed the existence of\r
infinitely many subharmonics both in the subquadratic and superquadratic\r
case, with suitable growth conditions on $H'$. Again the duality\r
approach enabled Clarke and Ekeland in \cite{clar:eke:2} to treat the\r
same problem in the convex-subquadratic case, with growth conditions on\r
$H$ only.\r
\r
Recently, Michalek and Tarantello (see \cite{mich:tar} and \cite{tar})\r
have obtained lower bound on the number of subharmonics of period $kT$,\r
based on symmetry considerations and on pinching estimates, as in\r
Sect.~5.2 of this article.\r
\r
%\r
% ---- Bibliography ----\r
%\r
\begin{thebibliography}{5}\r
%\r
\bibitem {clar:eke}\r
Clarke, F., Ekeland, I.:\r
Nonlinear oscillations and\r
boundary-value problems for Hamiltonian systems.\r
Arch. Rat. Mech. Anal. 78, 315--333 (1982)\r
\r
\bibitem {clar:eke:2}\r
Clarke, F., Ekeland, I.:\r
Solutions p\'{e}riodiques, du\r
p\'{e}riode donn\'{e}e, des \'{e}quations hamiltoniennes.\r
Note CRAS Paris 287, 1013--1015 (1978)\r
\r
\bibitem {mich:tar}\r
Michalek, R., Tarantello, G.:\r
Subharmonic solutions with prescribed minimal\r
period for nonautonomous Hamiltonian systems.\r
J. Diff. Eq. 72, 28--55 (1988)\r
\r
\bibitem {tar}\r
Tarantello, G.:\r
Subharmonic solutions for Hamiltonian\r
systems via a $\bbbz_{p}$ pseudoindex theory.\r
Annali di Matematica Pura (to appear)\r
\r
\bibitem {rab}\r
Rabinowitz, P.:\r
On subharmonic solutions of a Hamiltonian system.\r
Comm. Pure Appl. Math. 33, 609--633 (1980)\r
\r
\end{thebibliography}\r
\r
%\r
% second contribution with nearly identical text,\r
% slightly changed contribution head (all entries\r
% appear as defaults), and modified bibliography\r
%\r
\title{Hamiltonian Mechanics2}\r
\r
\author{Ivar Ekeland\inst{1} \and Roger Temam\inst{2}}\r
\r
\institute{Princeton University, Princeton NJ 08544, USA\r
\and\r
Universit\'{e} de Paris-Sud,\r
Laboratoire d'Analyse Num\'{e}rique, B\^{a}timent 425,\\\r
F-91405 Orsay Cedex, France}\r
\r
\maketitle\r
%\r
% Modify the bibliography environment to call for the author-year\r
% system. This is done normally with the citeauthoryear option\r
% for a particular contribution.\r
\makeatletter\r
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%\r
\begin{abstract}\r
The abstract should summarize the contents of the paper\r
using at least 70 and at most 150 words. It will be set in 9-point\r
font size and be inset 1.0 cm from the right and left margins.\r
There will be two blank lines before and after the Abstract. \dots\r
\keywords{graph transformations, convex geometry, lattice computations,\r
convex polygons, triangulations, discrete geometry}\r
\end{abstract}\r
%\r
\section{Fixed-Period Problems: The Sublinear Case}\r
%\r
With this chapter, the preliminaries are over, and we begin the search\r
for periodic solutions to Hamiltonian systems. All this will be done in\r
the convex case; that is, we shall study the boundary-value problem\r
\begin{eqnarray*}\r
  \dot{x}&=&JH' (t,x)\\\r
  x(0) &=& x(T)\r
\end{eqnarray*}\r
with $H(t,\cdot)$ a convex function of $x$, going to $+\infty$ when\r
$\left\|x\right\| \to \infty$.\r
\r
%\r
\subsection{Autonomous Systems}\r
%\r
In this section, we will consider the case when the Hamiltonian $H(x)$\r
is autonomous. For the sake of simplicity, we shall also assume that it\r
is $C^{1}$.\r
\r
We shall first consider the question of nontriviality, within the\r
general framework of\r
$\left(A_{\infty},B_{\infty}\right)$-subquadratic Hamiltonians. In\r
the second subsection, we shall look into the special case when $H$ is\r
$\left(0,b_{\infty}\right)$-subquadratic,\r
and we shall try to derive additional information.\r
%\r
\subsubsection{The General Case: Nontriviality.}\r
%\r
We assume that $H$ is\r
$\left(A_{\infty},B_{\infty}\right)$-sub\-qua\-dra\-tic at infinity,\r
for some constant symmetric matrices $A_{\infty}$ and $B_{\infty}$,\r
with $B_{\infty}-A_{\infty}$ positive definite. Set:\r
\begin{eqnarray}\r
\gamma :&=&{\rm smallest\ eigenvalue\ of}\ \ B_{\infty} - A_{\infty} \\\r
  \lambda : &=& {\rm largest\ negative\ eigenvalue\ of}\ \\r
  J \frac{d}{dt} +A_{\infty}\ .\r
\end{eqnarray}\r
\r
Theorem 21 tells us that if $\lambda +\gamma < 0$, the boundary-value\r
problem:\r
\begin{equation}\r
\begin{array}{rcl}\r
  \dot{x}&=&JH' (x)\\\r
  x(0)&=&x (T)\r
\end{array}\r
\end{equation}\r
has at least one solution\r
$\overline{x}$, which is found by minimizing the dual\r
action functional:\r
\begin{equation}\r
  \psi (u) = \int_{o}^{T} \left[\frac{1}{2}\r
  \left(\Lambda_{o}^{-1} u,u\right) + N^{\ast} (-u)\right] dt\r
\end{equation}\r
on the range of $\Lambda$, which is a subspace $R (\Lambda)_{L}^{2}$\r
with finite codimension. Here\r
\begin{equation}\r
  N(x) := H(x) - \frac{1}{2} \left(A_{\infty} x,x\right)\r
\end{equation}\r
is a convex function, and\r
\begin{equation}\r
  N(x) \le \frac{1}{2}\r
  \left(\left(B_{\infty} - A_{\infty}\right) x,x\right)\r
  + c\ \ \ \forall x\ .\r
\end{equation}\r
\r
%\r
\begin{proposition}\r
Assume $H'(0)=0$ and $ H(0)=0$. Set:\r
\begin{equation}\r
  \delta := \liminf_{x\to 0} 2 N (x) \left\|x\right\|^{-2}\ .\r
  \label{2eq:one}\r
\end{equation}\r
\r
If $\gamma < - \lambda < \delta$,\r
the solution $\overline{u}$ is non-zero:\r
\begin{equation}\r
  \overline{x} (t) \ne 0\ \ \ \forall t\ .\r
\end{equation}\r
\end{proposition}\r
%\r
\begin{proof}\r
Condition (\ref{2eq:one}) means that, for every\r
$\delta ' > \delta$, there is some $\varepsilon > 0$ such that\r
\begin{equation}\r
  \left\|x\right\| \le \varepsilon \Rightarrow N (x) \le\r
  \frac{\delta '}{2} \left\|x\right\|^{2}\ .\r
\end{equation}\r
\r
It is an exercise in convex analysis, into which we shall not go, to\r
show that this implies that there is an $\eta > 0$ such that\r
\begin{equation}\r
  f\left\|x\right\| \le \eta\r
  \Rightarrow N^{\ast} (y) \le \frac{1}{2\delta '}\r
  \left\|y\right\|^{2}\ .\r
  \label{2eq:two}\r
\end{equation}\r
\r
\begin{figure}\r
\vspace{2.5cm}\r
\caption{This is the caption of the figure displaying a white eagle and\r
a white horse on a snow field}\r
\end{figure}\r
\r
Since $u_{1}$ is a smooth function, we will have\r
$\left\|hu_{1}\right\|_\infty \le \eta$\r
for $h$ small enough, and inequality (\ref{2eq:two}) will hold,\r
yielding thereby:\r
\begin{equation}\r
  \psi (hu_{1}) \le \frac{h^{2}}{2}\r
  \frac{1}{\lambda} \left\|u_{1} \right\|_{2}^{2} + \frac{h^{2}}{2}\r
  \frac{1}{\delta '} \left\|u_{1}\right\|^{2}\ .\r
\end{equation}\r
\r
If we choose $\delta '$ close enough to $\delta$, the quantity\r
$\left(\frac{1}{\lambda} + \frac{1}{\delta '}\right)$\r
will be negative, and we end up with\r
\begin{equation}\r
  \psi (hu_{1}) < 0\ \ \ \ \ {\rm for}\ \ h\ne 0\ \ {\rm small}\ .\r
\end{equation}\r
\r
On the other hand, we check directly that $\psi (0) = 0$. This shows\r
that 0 cannot be a minimizer of $\psi$, not even a local one.\r
So $\overline{u} \ne 0$ and\r
$\overline{u} \ne \Lambda_{o}^{-1} (0) = 0$. \qed\r
\end{proof}\r
%\r
\begin{corollary}\r
Assume $H$ is $C^{2}$ and\r
$\left(a_{\infty},b_{\infty}\right)$-subquadratic at infinity. Let\r
$\xi_{1},\allowbreak\dots,\allowbreak\xi_{N}$  be the\r
equilibria, that is, the solutions of $H' (\xi ) = 0$.\r
Denote by $\omega_{k}$\r
the smallest eigenvalue of $H'' \left(\xi_{k}\right)$, and set:\r
\begin{equation}\r
  \omega : = {\rm Min\,} \left\{\omega_{1},\dots,\omega_{k}\right\}\ .\r
\end{equation}\r
If:\r
\begin{equation}\r
  \frac{T}{2\pi} b_{\infty} <\r
  - E \left[- \frac{T}{2\pi}a_{\infty}\right] <\r
  \frac{T}{2\pi}\omega\r
  \label{2eq:three}\r
\end{equation}\r
then minimization of $\psi$ yields a non-constant $T$-periodic solution\r
$\overline{x}$.\r
\end{corollary}\r
%\r
\r
We recall once more that by the integer part $E [\alpha ]$ of\r
$\alpha \in \bbbr$, we mean the $a\in \bbbz$\r
such that $a< \alpha \le a+1$. For instance,\r
if we take $a_{\infty} = 0$, Corollary 2 tells\r
us that $\overline{x}$ exists and is\r
non-constant provided that:\r
\r
\begin{equation}\r
  \frac{T}{2\pi} b_{\infty} < 1 < \frac{T}{2\pi}\r
\end{equation}\r
or\r
\begin{equation}\r
  T\in \left(\frac{2\pi}{\omega},\frac{2\pi}{b_{\infty}}\right)\ .\r
  \label{2eq:four}\r
\end{equation}\r
\r
%\r
\begin{proof}\r
The spectrum of $\Lambda$ is $\frac{2\pi}{T} \bbbz +a_{\infty}$. The\r
largest negative eigenvalue $\lambda$ is given by\r
$\frac{2\pi}{T}k_{o} +a_{\infty}$,\r
where\r
\begin{equation}\r
  \frac{2\pi}{T}k_{o} + a_{\infty} < 0\r
  \le \frac{2\pi}{T} (k_{o} +1) + a_{\infty}\ .\r
\end{equation}\r
Hence:\r
\begin{equation}\r
  k_{o} = E \left[- \frac{T}{2\pi} a_{\infty}\right] \ .\r
\end{equation}\r
\r
The condition $\gamma < -\lambda < \delta$ now becomes:\r
\begin{equation}\r
  b_{\infty} - a_{\infty} <\r
  - \frac{2\pi}{T} k_{o} -a_{\infty} < \omega -a_{\infty}\r
\end{equation}\r
which is precisely condition (\ref{2eq:three}).\qed\r
\end{proof}\r
%\r
\r
\begin{lemma}\r
Assume that $H$ is $C^{2}$ on $\bbbr^{2n} \setminus \{ 0\}$ and\r
that $H'' (x)$ is non-de\-gen\-er\-ate for any $x\ne 0$. Then any local\r
minimizer $\widetilde{x}$ of $\psi$ has minimal period $T$.\r
\end{lemma}\r
%\r
\begin{proof}\r
We know that $\widetilde{x}$, or\r
$\widetilde{x} + \xi$ for some constant $\xi\r
\in \bbbr^{2n}$, is a $T$-periodic solution of the Hamiltonian system:\r
\begin{equation}\r
  \dot{x} = JH' (x)\ .\r
\end{equation}\r
\r
There is no loss of generality in taking $\xi = 0$. So\r
$\psi (x) \ge \psi (\widetilde{x} )$\r
for all $\widetilde{x}$ in some neighbourhood of $x$ in\r
$W^{1,2} \left(\bbbr / T\bbbz ; \bbbr^{2n}\right)$.\r
\r
But this index is precisely the index\r
$i_{T} (\widetilde{x} )$ of the $T$-periodic\r
solution $\widetilde{x}$ over the interval\r
$(0,T)$, as defined in Sect.~2.6. So\r
\begin{equation}\r
  i_{T} (\widetilde{x} ) = 0\ .\r
  \label{2eq:five}\r
\end{equation}\r
\r
Now if $\widetilde{x}$ has a lower period, $T/k$ say,\r
we would have, by Corollary 31:\r
\begin{equation}\r
  i_{T} (\widetilde{x} ) =\r
  i_{kT/k}(\widetilde{x} ) \ge\r
  ki_{T/k} (\widetilde{x} ) + k-1 \ge k-1 \ge 1\ .\r
\end{equation}\r
\r
This would contradict (\ref{2eq:five}), and thus cannot happen.\qed\r
\end{proof}\r
%\r
\paragraph{Notes and Comments.}\r
The results in this section are a\r
refined version of \cite{2clar:eke};\r
the minimality result of Proposition\r
14 was the first of its kind.\r
\r
To understand the nontriviality conditions, such as the one in formula\r
(\ref{2eq:four}), one may think of a one-parameter family\r
$x_{T}$, $T\in \left(2\pi\omega^{-1}, 2\pi b_{\infty}^{-1}\right)$\r
of periodic solutions, $x_{T} (0) = x_{T} (T)$,\r
with $x_{T}$ going away to infinity when $T\to 2\pi \omega^{-1}$,\r
which is the period of the linearized system at 0.\r
\r
\begin{table}\r
\caption{This is the example table taken out of {\it The\r
\TeX{}book,} p.\,246}\r
\begin{center}\r
\begin{tabular}{r@{\quad}rl}\r
\hline\r
\multicolumn{1}{l}{\rule{0pt}{12pt}\r
                   Year}&\multicolumn{2}{l}{World population}\\[2pt]\r
\hline\rule{0pt}{12pt}\r
8000 B.C.  &     5,000,000& \\\r
  50 A.D.  &   200,000,000& \\\r
1650 A.D.  &   500,000,000& \\\r
1945 A.D.  & 2,300,000,000& \\\r
1980 A.D.  & 4,400,000,000& \\[2pt]\r
\hline\r
\end{tabular}\r
\end{center}\r
\end{table}\r
%\r
\begin{theorem} [Ghoussoub-Preiss]\r
Assume $H(t,x)$ is\r
$(0,\varepsilon )$-subquadratic at\r
infinity for all $\varepsilon > 0$, and $T$-periodic in $t$\r
\begin{equation}\r
  H (t,\cdot )\ \ \ \ \ {\rm is\ convex}\ \ \forall t\r
\end{equation}\r
\begin{equation}\r
  H (\cdot ,x)\ \ \ \ \ {\rm is}\ \ T{\rm -periodic}\ \ \forall x\r
\end{equation}\r
\begin{equation}\r
  H (t,x)\ge n\left(\left\|x\right\|\right)\ \ \ \ \\r
  {\rm with}\ \ n (s)s^{-1}\to \infty\ \ {\rm as}\ \ s\to \infty\r
\end{equation}\r
\begin{equation}\r
  \forall \varepsilon > 0\ ,\ \ \ \exists c\ :\\r
  H(t,x) \le \frac{\varepsilon}{2}\left\|x\right\|^{2} + c\ .\r
\end{equation}\r
\r
Assume also that $H$ is $C^{2}$, and $H'' (t,x)$ is positive definite\r
everywhere. Then there is a sequence $x_{k}$, $k\in \bbbn$, of\r
$kT$-periodic solutions of the system\r
\begin{equation}\r
  \dot{x} = JH' (t,x)\r
\end{equation}\r
such that, for every $k\in \bbbn$, there is some $p_{o}\in\bbbn$ with:\r
\begin{equation}\r
  p\ge p_{o}\Rightarrow x_{pk} \ne x_{k}\ .\r
\end{equation}\r
\qed\r
\end{theorem}\r
%\r
\begin{example} [{{\rm External forcing}}]\r
Consider the system:\r
\begin{equation}\r
  \dot{x} = JH' (x) + f(t)\r
\end{equation}\r
where the Hamiltonian $H$ is\r
$\left(0,b_{\infty}\right)$-subquadratic, and the\r
forcing term is a distribution on the circle:\r
\begin{equation}\r
  f = \frac{d}{dt} F + f_{o}\ \ \ \ \\r
  {\rm with}\ \ F\in L^{2} \left(\bbbr / T\bbbz; \bbbr^{2n}\right)\ ,\r
\end{equation}\r
where $f_{o} : = T^{-1}\int_{o}^{T} f (t) dt$. For instance,\r
\begin{equation}\r
  f (t) = \sum_{k\in \bbbn} \delta_{k} \xi\ ,\r
\end{equation}\r
where $\delta_{k}$ is the Dirac mass at $t= k$ and\r
$\xi \in \bbbr^{2n}$ is a\r
constant, fits the prescription. This means that the system\r
$\dot{x} = JH' (x)$ is being excited by a\r
series of identical shocks at interval $T$.\r
\end{example}\r
%\r
\begin{definition}\r
Let $A_{\infty} (t)$ and $B_{\infty} (t)$ be symmetric\r
operators in $\bbbr^{2n}$, depending continuously on\r
$t\in [0,T]$, such that\r
$A_{\infty} (t) \le B_{\infty} (t)$ for all $t$.\r
\r
A Borelian function\r
$H: [0,T]\times \bbbr^{2n} \to \bbbr$\r
is called\r
$\left(A_{\infty} ,B_{\infty}\right)$-{\it subquadratic at infinity}\r
if there exists a function $N(t,x)$ such that:\r
\begin{equation}\r
  H (t,x) = \frac{1}{2} \left(A_{\infty} (t) x,x\right) + N(t,x)\r
\end{equation}\r
\begin{equation}\r
  \forall t\ ,\ \ \ N(t,x)\ \ \ \ \\r
  {\rm is\ convex\ with\  respect\  to}\ \ x\r
\end{equation}\r
\begin{equation}\r
  N(t,x) \ge n\left(\left\|x\right\|\right)\ \ \ \ \\r
  {\rm with}\ \ n(s)s^{-1}\to +\infty\ \ {\rm as}\ \ s\to +\infty\r
\end{equation}\r
\begin{equation}\r
  \exists c\in \bbbr\ :\ \ \ H (t,x) \le\r
  \frac{1}{2} \left(B_{\infty} (t) x,x\right) + c\ \ \ \forall x\ .\r
\end{equation}\r
\r
If $A_{\infty} (t) = a_{\infty} I$ and\r
$B_{\infty} (t) = b_{\infty} I$, with\r
$a_{\infty} \le b_{\infty} \in \bbbr$,\r
we shall say that $H$ is\r
$\left(a_{\infty},b_{\infty}\right)$-subquadratic\r
at infinity. As an example, the function\r
$\left\|x\right\|^{\alpha}$, with\r
$1\le \alpha < 2$, is $(0,\varepsilon )$-subquadratic at infinity\r
for every $\varepsilon > 0$. Similarly, the Hamiltonian\r
\begin{equation}\r
H (t,x) = \frac{1}{2} k \left\|k\right\|^{2} +\left\|x\right\|^{\alpha}\r
\end{equation}\r
is $(k,k+\varepsilon )$-subquadratic for every $\varepsilon > 0$.\r
Note that, if $k<0$, it is not convex.\r
\end{definition}\r
%\r
\r
\paragraph{Notes and Comments.}\r
The first results on subharmonics were\r
obtained by Rabinowitz in \cite{2rab}, who showed the existence of\r
infinitely many subharmonics both in the subquadratic and superquadratic\r
case, with suitable growth conditions on $H'$. Again the duality\r
approach enabled Clarke and Ekeland in \cite{2clar:eke:2} to treat the\r
same problem in the convex-subquadratic case, with growth conditions on\r
$H$ only.\r
\r
Recently, Michalek and Tarantello (see Michalek, R., Tarantello, G.\r
\cite{2mich:tar} and Tarantello, G. \cite{2tar}) have obtained lower\r
bound on the number of subharmonics of period $kT$, based on symmetry\r
considerations and on pinching estimates, as in Sect.~5.2 of this\r
article.\r
\r
%\r
% ---- Bibliography ----\r
%\r
\begin{thebibliography}{}\r
%\r
\bibitem[1980]{2clar:eke}\r
Clarke, F., Ekeland, I.:\r
Nonlinear oscillations and\r
boundary-value problems for Hamiltonian systems.\r
Arch. Rat. Mech. Anal. 78, 315--333 (1982)\r
\r
\bibitem[1981]{2clar:eke:2}\r
Clarke, F., Ekeland, I.:\r
Solutions p\'{e}riodiques, du\r
p\'{e}riode donn\'{e}e, des \'{e}quations hamiltoniennes.\r
Note CRAS Paris 287, 1013--1015 (1978)\r
\r
\bibitem[1982]{2mich:tar}\r
Michalek, R., Tarantello, G.:\r
Subharmonic solutions with prescribed minimal\r
period for nonautonomous Hamiltonian systems.\r
J. Diff. Eq. 72, 28--55 (1988)\r
\r
\bibitem[1983]{2tar}\r
Tarantello, G.:\r
Subharmonic solutions for Hamiltonian\r
systems via a $\bbbz_{p}$ pseudoindex theory.\r
Annali di Matematica Pura (to appear)\r
\r
\bibitem[1985]{2rab}\r
Rabinowitz, P.:\r
On subharmonic solutions of a Hamiltonian system.\r
Comm. Pure Appl. Math. 33, 609--633 (1980)\r
\r
\end{thebibliography}\r
\clearpage\r
\addtocmark[2]{Author Index} % additional numbered TOC entry\r
\renewcommand{\indexname}{Author Index}\r
\printindex\r
\clearpage\r
\addtocmark[2]{Subject Index} % additional numbered TOC entry\r
\markboth{Subject Index}{Subject Index}\r
\renewcommand{\indexname}{Subject Index}\r
\input{subjidx.ind}\r
\end{document}\r