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\documentclass{llncs} | |
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\usepackage{makeidx} % allows for indexgeneration | |
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\begin{document} | |
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\frontmatter % for the preliminaries | |
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\pagestyle{headings} % switches on printing of running heads | |
\addtocmark{Hamiltonian Mechanics} % additional mark in the TOC | |
% | |
\chapter*{Preface} | |
% | |
This textbook is intended for use by students of physics, physical | |
chemistry, and theoretical chemistry. The reader is presumed to have a | |
basic knowledge of atomic and quantum physics at the level provided, for | |
example, by the first few chapters in our book {\it The Physics of Atoms | |
and Quanta}. The student of physics will find here material which should | |
be included in the basic education of every physicist. This book should | |
furthermore allow students to acquire an appreciation of the breadth and | |
variety within the field of molecular physics and its future as a | |
fascinating area of research. | |
For the student of chemistry, the concepts introduced in this book will | |
provide a theoretical framework for that entire field of study. With the | |
help of these concepts, it is at least in principle possible to reduce | |
the enormous body of empirical chemical knowledge to a few basic | |
principles: those of quantum mechanics. In addition, modern physical | |
methods whose fundamentals are introduced here are becoming increasingly | |
important in chemistry and now represent indispensable tools for the | |
chemist. As examples, we might mention the structural analysis of | |
complex organic compounds, spectroscopic investigation of very rapid | |
reaction processes or, as a practical application, the remote detection | |
of pollutants in the air. | |
\vspace{1cm} | |
\begin{flushright}\noindent | |
April 1995\hfill Walter Olthoff\\ | |
Program Chair\\ | |
ECOOP'95 | |
\end{flushright} | |
% | |
\chapter*{Organization} | |
ECOOP'95 is organized by the department of Computer Science, Univeristy | |
of \AA rhus and AITO (association Internationa pour les Technologie | |
Object) in cooperation with ACM/SIGPLAN. | |
% | |
\section*{Executive Commitee} | |
\begin{tabular}{@{}p{5cm}@{}p{7.2cm}@{}} | |
Conference Chair:&Ole Lehrmann Madsen (\AA rhus University, DK)\\ | |
Program Chair: &Walter Olthoff (DFKI GmbH, Germany)\\ | |
Organizing Chair:&J\o rgen Lindskov Knudsen (\AA rhus University, DK)\\ | |
Tutorials:&Birger M\o ller-Pedersen\hfil\break | |
(Norwegian Computing Center, Norway)\\ | |
Workshops:&Eric Jul (University of Kopenhagen, Denmark)\\ | |
Panels:&Boris Magnusson (Lund University, Sweden)\\ | |
Exhibition:&Elmer Sandvad (\AA rhus University, DK)\\ | |
Demonstrations:&Kurt N\o rdmark (\AA rhus University, DK) | |
\end{tabular} | |
% | |
\section*{Program Commitee} | |
\begin{tabular}{@{}p{5cm}@{}p{7.2cm}@{}} | |
Conference Chair:&Ole Lehrmann Madsen (\AA rhus University, DK)\\ | |
Program Chair: &Walter Olthoff (DFKI GmbH, Germany)\\ | |
Organizing Chair:&J\o rgen Lindskov Knudsen (\AA rhus University, DK)\\ | |
Tutorials:&Birger M\o ller-Pedersen\hfil\break | |
(Norwegian Computing Center, Norway)\\ | |
Workshops:&Eric Jul (University of Kopenhagen, Denmark)\\ | |
Panels:&Boris Magnusson (Lund University, Sweden)\\ | |
Exhibition:&Elmer Sandvad (\AA rhus University, DK)\\ | |
Demonstrations:&Kurt N\o rdmark (\AA rhus University, DK) | |
\end{tabular} | |
% | |
\begin{multicols}{3}[\section*{Referees}] | |
V.~Andreev\\ | |
B\"arwolff\\ | |
E.~Barrelet\\ | |
H.P.~Beck\\ | |
G.~Bernardi\\ | |
E.~Binder\\ | |
P.C.~Bosetti\\ | |
Braunschweig\\ | |
F.W.~B\"usser\\ | |
T.~Carli\\ | |
A.B.~Clegg\\ | |
G.~Cozzika\\ | |
S.~Dagoret\\ | |
Del~Buono\\ | |
P.~Dingus\\ | |
H.~Duhm\\ | |
J.~Ebert\\ | |
S.~Eichenberger\\ | |
R.J.~Ellison\\ | |
Feltesse\\ | |
W.~Flauger\\ | |
A.~Fomenko\\ | |
G.~Franke\\ | |
J.~Garvey\\ | |
M.~Gennis\\ | |
L.~Goerlich\\ | |
P.~Goritchev\\ | |
H.~Greif\\ | |
E.M.~Hanlon\\ | |
R.~Haydar\\ | |
R.C.W.~Henderso\\ | |
P.~Hill\\ | |
H.~Hufnagel\\ | |
A.~Jacholkowska\\ | |
Johannsen\\ | |
S.~Kasarian\\ | |
I.R.~Kenyon\\ | |
C.~Kleinwort\\ | |
T.~K\"ohler\\ | |
S.D.~Kolya\\ | |
P.~Kostka\\ | |
U.~Kr\"uger\\ | |
J.~Kurzh\"ofer\\ | |
M.P.J.~Landon\\ | |
A.~Lebedev\\ | |
Ch.~Ley\\ | |
F.~Linsel\\ | |
H.~Lohmand\\ | |
Martin\\ | |
S.~Masson\\ | |
K.~Meier\\ | |
C.A.~Meyer\\ | |
S.~Mikocki\\ | |
J.V.~Morris\\ | |
B.~Naroska\\ | |
Nguyen\\ | |
U.~Obrock\\ | |
G.D.~Patel\\ | |
Ch.~Pichler\\ | |
S.~Prell\\ | |
F.~Raupach\\ | |
V.~Riech\\ | |
P.~Robmann\\ | |
N.~Sahlmann\\ | |
P.~Schleper\\ | |
Sch\"oning\\ | |
B.~Schwab\\ | |
A.~Semenov\\ | |
G.~Siegmon\\ | |
J.R.~Smith\\ | |
M.~Steenbock\\ | |
U.~Straumann\\ | |
C.~Thiebaux\\ | |
P.~Van~Esch\\ | |
from Yerevan Ph\\ | |
L.R.~West\\ | |
G.-G.~Winter\\ | |
T.P.~Yiou\\ | |
M.~Zimmer\end{multicols} | |
% | |
\section*{Sponsoring Institutions} | |
% | |
Bernauer-Budiman Inc., Reading, Mass.\\ | |
The Hofmann-International Company, San Louis Obispo, Cal.\\ | |
Kramer Industries, Heidelberg, Germany | |
% | |
\tableofcontents | |
% | |
\mainmatter % start of the contributions | |
% | |
\title{Hamiltonian Mechanics unter besonderer Ber\"ucksichtigung der | |
h\"ohreren Lehranstalten} | |
% | |
\titlerunning{Hamiltonian Mechanics} % abbreviated title (for running head) | |
% also used for the TOC unless | |
% \toctitle is used | |
% | |
\author{Ivar Ekeland\inst{1} \and Roger Temam\inst{2} | |
Jeffrey Dean \and David Grove \and Craig Chambers \and Kim~B.~Bruce \and | |
Elsa Bertino} | |
% | |
\authorrunning{Ivar Ekeland et al.} % abbreviated author list (for running head) | |
% | |
%%%% list of authors for the TOC (use if author list has to be modified) | |
\tocauthor{Ivar Ekeland, Roger Temam, Jeffrey Dean, David Grove, | |
Craig Chambers, Kim B. Bruce, and Elisa Bertino} | |
% | |
\institute{Princeton University, Princeton NJ 08544, USA,\\ | |
\email{I.Ekeland@princeton.edu},\\ WWW home page: | |
\texttt{http://users/\homedir iekeland/web/welcome.html} | |
\and | |
Universit\'{e} de Paris-Sud, | |
Laboratoire d'Analyse Num\'{e}rique, B\^{a}timent 425,\\ | |
F-91405 Orsay Cedex, France} | |
\maketitle % typeset the title of the contribution | |
\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 | |
\keywords{computational geometry, graph theory, Hamilton cycles} | |
\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~\ref{ghou:pre} tells us that if $\lambda +\gamma < 0$, the | |
boundary-value problem: | |
\begin{equation} | |
\begin{array}{rcl} | |
\dot{x}&=&JH' (x)\\ | |
x(0)&=&x (T) | |
\end{array} | |
\end{equation} | |
has at least one solution | |
$\overline{x}$, which is found by minimizing the dual | |
action functional: | |
\begin{equation} | |
\psi (u) = \int_{o}^{T} \left[\frac{1}{2} | |
\left(\Lambda_{o}^{-1} u,u\right) + N^{\ast} (-u)\right] dt | |
\end{equation} | |
on the range of $\Lambda$, which is a subspace $R (\Lambda)_{L}^{2}$ | |
with finite codimension. Here | |
\begin{equation} | |
N(x) := H(x) - \frac{1}{2} \left(A_{\infty} x,x\right) | |
\end{equation} | |
is a convex function, and | |
\begin{equation} | |
N(x) \le \frac{1}{2} | |
\left(\left(B_{\infty} - A_{\infty}\right) x,x\right) | |
+ c\ \ \ \forall x\ . | |
\end{equation} | |
% | |
\begin{proposition} | |
Assume $H'(0)=0$ and $ H(0)=0$. Set: | |
\begin{equation} | |
\delta := \liminf_{x\to 0} 2 N (x) \left\|x\right\|^{-2}\ . | |
\label{eq:one} | |
\end{equation} | |
If $\gamma < - \lambda < \delta$, | |
the solution $\overline{u}$ is non-zero: | |
\begin{equation} | |
\overline{x} (t) \ne 0\ \ \ \forall t\ . | |
\end{equation} | |
\end{proposition} | |
% | |
\begin{proof} | |
Condition (\ref{eq:one}) means that, for every | |
$\delta ' > \delta$, there is some $\varepsilon > 0$ such that | |
\begin{equation} | |
\left\|x\right\| \le \varepsilon \Rightarrow N (x) \le | |
\frac{\delta '}{2} \left\|x\right\|^{2}\ . | |
\end{equation} | |
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 | |
\begin{equation} | |
f\left\|x\right\| \le \eta | |
\Rightarrow N^{\ast} (y) \le \frac{1}{2\delta '} | |
\left\|y\right\|^{2}\ . | |
\label{eq:two} | |
\end{equation} | |
\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: | |
\begin{equation} | |
\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}\ . | |
\end{equation} | |
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 | |
\begin{equation} | |
\psi (hu_{1}) < 0\ \ \ \ \ {\rm for}\ \ h\ne 0\ \ {\rm small}\ . | |
\end{equation} | |
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: | |
\begin{equation} | |
\omega : = {\rm Min\,} \left\{\omega_{1},\dots,\omega_{k}\right\}\ . | |
\end{equation} | |
If: | |
\begin{equation} | |
\frac{T}{2\pi} b_{\infty} < | |
- E \left[- \frac{T}{2\pi}a_{\infty}\right] < | |
\frac{T}{2\pi}\omega | |
\label{eq:three} | |
\end{equation} | |
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: | |
\begin{equation} | |
\frac{T}{2\pi} b_{\infty} < 1 < \frac{T}{2\pi} | |
\end{equation} | |
or | |
\begin{equation} | |
T\in \left(\frac{2\pi}{\omega},\frac{2\pi}{b_{\infty}}\right)\ . | |
\label{eq:four} | |
\end{equation} | |
% | |
\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 | |
\begin{equation} | |
\frac{2\pi}{T}k_{o} + a_{\infty} < 0 | |
\le \frac{2\pi}{T} (k_{o} +1) + a_{\infty}\ . | |
\end{equation} | |
Hence: | |
\begin{equation} | |
k_{o} = E \left[- \frac{T}{2\pi} a_{\infty}\right] \ . | |
\end{equation} | |
The condition $\gamma < -\lambda < \delta$ now becomes: | |
\begin{equation} | |
b_{\infty} - a_{\infty} < | |
- \frac{2\pi}{T} k_{o} -a_{\infty} < \omega -a_{\infty} | |
\end{equation} | |
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: | |
\begin{equation} | |
\dot{x} = JH' (x)\ . | |
\end{equation} | |
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 | |
\begin{equation} | |
i_{T} (\widetilde{x} ) = 0\ . | |
\label{eq:five} | |
\end{equation} | |
Now if $\widetilde{x}$ has a lower period, $T/k$ say, | |
we would have, by Corollary 31: | |
\begin{equation} | |
i_{T} (\widetilde{x} ) = | |
i_{kT/k}(\widetilde{x} ) \ge | |
ki_{T/k} (\widetilde{x} ) + k-1 \ge k-1 \ge 1\ . | |
\end{equation} | |
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} | |
\begin{center} | |
\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{center} | |
\end{table} | |
% | |
\begin{theorem} [Ghoussoub-Preiss]\label{ghou:pre} | |
Assume $H(t,x)$ is | |
$(0,\varepsilon )$-subquadratic at | |
infinity for all $\varepsilon > 0$, and $T$-periodic in $t$ | |
\begin{equation} | |
H (t,\cdot )\ \ \ \ \ {\rm is\ convex}\ \ \forall t | |
\end{equation} | |
\begin{equation} | |
H (\cdot ,x)\ \ \ \ \ {\rm is}\ \ T{\rm -periodic}\ \ \forall x | |
\end{equation} | |
\begin{equation} | |
H (t,x)\ge n\left(\left\|x\right\|\right)\ \ \ \ \ | |
{\rm with}\ \ n (s)s^{-1}\to \infty\ \ {\rm as}\ \ s\to \infty | |
\end{equation} | |
\begin{equation} | |
\forall \varepsilon > 0\ ,\ \ \ \exists c\ :\ | |
H(t,x) \le \frac{\varepsilon}{2}\left\|x\right\|^{2} + c\ . | |
\end{equation} | |
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 | |
\begin{equation} | |
\dot{x} = JH' (t,x) | |
\end{equation} | |
such that, for every $k\in \bbbn$, there is some $p_{o}\in\bbbn$ with: | |
\begin{equation} | |
p\ge p_{o}\Rightarrow x_{pk} \ne x_{k}\ . | |
\end{equation} | |
\qed | |
\end{theorem} | |
% | |
\begin{example} [{{\rm External forcing}}] | |
Consider the system: | |
\begin{equation} | |
\dot{x} = JH' (x) + f(t) | |
\end{equation} | |
where the Hamiltonian $H$ is | |
$\left(0,b_{\infty}\right)$-subquadratic, and the | |
forcing term is a distribution on the circle: | |
\begin{equation} | |
f = \frac{d}{dt} F + f_{o}\ \ \ \ \ | |
{\rm with}\ \ F\in L^{2} \left(\bbbr / T\bbbz; \bbbr^{2n}\right)\ , | |
\end{equation} | |
where $f_{o} : = T^{-1}\int_{o}^{T} f (t) dt$. For instance, | |
\begin{equation} | |
f (t) = \sum_{k\in \bbbn} \delta_{k} \xi\ , | |
\end{equation} | |
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: | |
\begin{equation} | |
H (t,x) = \frac{1}{2} \left(A_{\infty} (t) x,x\right) + N(t,x) | |
\end{equation} | |
\begin{equation} | |
\forall t\ ,\ \ \ N(t,x)\ \ \ \ \ | |
{\rm is\ convex\ with\ respect\ to}\ \ x | |
\end{equation} | |
\begin{equation} | |
N(t,x) \ge n\left(\left\|x\right\|\right)\ \ \ \ \ | |
{\rm with}\ \ n(s)s^{-1}\to +\infty\ \ {\rm as}\ \ s\to +\infty | |
\end{equation} | |
\begin{equation} | |
\exists c\in \bbbr\ :\ \ \ H (t,x) \le | |
\frac{1}{2} \left(B_{\infty} (t) x,x\right) + c\ \ \ \forall x\ . | |
\end{equation} | |
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 | |
\begin{equation} | |
H (t,x) = \frac{1}{2} k \left\|k\right\|^{2} +\left\|x\right\|^{\alpha} | |
\end{equation} | |
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. 78, 315--333 (1982) | |
\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 287, 1013--1015 (1978) | |
\bibitem {mich:tar} | |
Michalek, R., Tarantello, G.: | |
Subharmonic solutions with prescribed minimal | |
period for nonautonomous Hamiltonian systems. | |
J. Diff. Eq. 72, 28--55 (1988) | |
\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. 33, 609--633 (1980) | |
\end{thebibliography} | |
% | |
% second contribution with nearly identical text, | |
% slightly changed contribution head (all entries | |
% appear as defaults), and modified bibliography | |
% | |
\title{Hamiltonian Mechanics2} | |
\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 | |
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% | |
\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 | |
\keywords{graph transformations, convex geometry, lattice computations, | |
convex polygons, triangulations, discrete geometry} | |
\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{equation} | |
\begin{array}{rcl} | |
\dot{x}&=&JH' (x)\\ | |
x(0)&=&x (T) | |
\end{array} | |
\end{equation} | |
has at least one solution | |
$\overline{x}$, which is found by minimizing the dual | |
action functional: | |
\begin{equation} | |
\psi (u) = \int_{o}^{T} \left[\frac{1}{2} | |
\left(\Lambda_{o}^{-1} u,u\right) + N^{\ast} (-u)\right] dt | |
\end{equation} | |
on the range of $\Lambda$, which is a subspace $R (\Lambda)_{L}^{2}$ | |
with finite codimension. Here | |
\begin{equation} | |
N(x) := H(x) - \frac{1}{2} \left(A_{\infty} x,x\right) | |
\end{equation} | |
is a convex function, and | |
\begin{equation} | |
N(x) \le \frac{1}{2} | |
\left(\left(B_{\infty} - A_{\infty}\right) x,x\right) | |
+ c\ \ \ \forall x\ . | |
\end{equation} | |
% | |
\begin{proposition} | |
Assume $H'(0)=0$ and $ H(0)=0$. Set: | |
\begin{equation} | |
\delta := \liminf_{x\to 0} 2 N (x) \left\|x\right\|^{-2}\ . | |
\label{2eq:one} | |
\end{equation} | |
If $\gamma < - \lambda < \delta$, | |
the solution $\overline{u}$ is non-zero: | |
\begin{equation} | |
\overline{x} (t) \ne 0\ \ \ \forall t\ . | |
\end{equation} | |
\end{proposition} | |
% | |
\begin{proof} | |
Condition (\ref{2eq:one}) means that, for every | |
$\delta ' > \delta$, there is some $\varepsilon > 0$ such that | |
\begin{equation} | |
\left\|x\right\| \le \varepsilon \Rightarrow N (x) \le | |
\frac{\delta '}{2} \left\|x\right\|^{2}\ . | |
\end{equation} | |
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 | |
\begin{equation} | |
f\left\|x\right\| \le \eta | |
\Rightarrow N^{\ast} (y) \le \frac{1}{2\delta '} | |
\left\|y\right\|^{2}\ . | |
\label{2eq:two} | |
\end{equation} | |
\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{2eq:two}) will hold, | |
yielding thereby: | |
\begin{equation} | |
\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}\ . | |
\end{equation} | |
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 | |
\begin{equation} | |
\psi (hu_{1}) < 0\ \ \ \ \ {\rm for}\ \ h\ne 0\ \ {\rm small}\ . | |
\end{equation} | |
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: | |
\begin{equation} | |
\omega : = {\rm Min\,} \left\{\omega_{1},\dots,\omega_{k}\right\}\ . | |
\end{equation} | |
If: | |
\begin{equation} | |
\frac{T}{2\pi} b_{\infty} < | |
- E \left[- \frac{T}{2\pi}a_{\infty}\right] < | |
\frac{T}{2\pi}\omega | |
\label{2eq:three} | |
\end{equation} | |
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: | |
\begin{equation} | |
\frac{T}{2\pi} b_{\infty} < 1 < \frac{T}{2\pi} | |
\end{equation} | |
or | |
\begin{equation} | |
T\in \left(\frac{2\pi}{\omega},\frac{2\pi}{b_{\infty}}\right)\ . | |
\label{2eq:four} | |
\end{equation} | |
% | |
\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 | |
\begin{equation} | |
\frac{2\pi}{T}k_{o} + a_{\infty} < 0 | |
\le \frac{2\pi}{T} (k_{o} +1) + a_{\infty}\ . | |
\end{equation} | |
Hence: | |
\begin{equation} | |
k_{o} = E \left[- \frac{T}{2\pi} a_{\infty}\right] \ . | |
\end{equation} | |
The condition $\gamma < -\lambda < \delta$ now becomes: | |
\begin{equation} | |
b_{\infty} - a_{\infty} < | |
- \frac{2\pi}{T} k_{o} -a_{\infty} < \omega -a_{\infty} | |
\end{equation} | |
which is precisely condition (\ref{2eq: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: | |
\begin{equation} | |
\dot{x} = JH' (x)\ . | |
\end{equation} | |
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 | |
\begin{equation} | |
i_{T} (\widetilde{x} ) = 0\ . | |
\label{2eq:five} | |
\end{equation} | |
Now if $\widetilde{x}$ has a lower period, $T/k$ say, | |
we would have, by Corollary 31: | |
\begin{equation} | |
i_{T} (\widetilde{x} ) = | |
i_{kT/k}(\widetilde{x} ) \ge | |
ki_{T/k} (\widetilde{x} ) + k-1 \ge k-1 \ge 1\ . | |
\end{equation} | |
This would contradict (\ref{2eq:five}), and thus cannot happen.\qed | |
\end{proof} | |
% | |
\paragraph{Notes and Comments.} | |
The results in this section are a | |
refined version of \cite{2clar: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{2eq: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} | |
\begin{center} | |
\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{center} | |
\end{table} | |
% | |
\begin{theorem} [Ghoussoub-Preiss] | |
Assume $H(t,x)$ is | |
$(0,\varepsilon )$-subquadratic at | |
infinity for all $\varepsilon > 0$, and $T$-periodic in $t$ | |
\begin{equation} | |
H (t,\cdot )\ \ \ \ \ {\rm is\ convex}\ \ \forall t | |
\end{equation} | |
\begin{equation} | |
H (\cdot ,x)\ \ \ \ \ {\rm is}\ \ T{\rm -periodic}\ \ \forall x | |
\end{equation} | |
\begin{equation} | |
H (t,x)\ge n\left(\left\|x\right\|\right)\ \ \ \ \ | |
{\rm with}\ \ n (s)s^{-1}\to \infty\ \ {\rm as}\ \ s\to \infty | |
\end{equation} | |
\begin{equation} | |
\forall \varepsilon > 0\ ,\ \ \ \exists c\ :\ | |
H(t,x) \le \frac{\varepsilon}{2}\left\|x\right\|^{2} + c\ . | |
\end{equation} | |
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 | |
\begin{equation} | |
\dot{x} = JH' (t,x) | |
\end{equation} | |
such that, for every $k\in \bbbn$, there is some $p_{o}\in\bbbn$ with: | |
\begin{equation} | |
p\ge p_{o}\Rightarrow x_{pk} \ne x_{k}\ . | |
\end{equation} | |
\qed | |
\end{theorem} | |
% | |
\begin{example} [{{\rm External forcing}}] | |
Consider the system: | |
\begin{equation} | |
\dot{x} = JH' (x) + f(t) | |
\end{equation} | |
where the Hamiltonian $H$ is | |
$\left(0,b_{\infty}\right)$-subquadratic, and the | |
forcing term is a distribution on the circle: | |
\begin{equation} | |
f = \frac{d}{dt} F + f_{o}\ \ \ \ \ | |
{\rm with}\ \ F\in L^{2} \left(\bbbr / T\bbbz; \bbbr^{2n}\right)\ , | |
\end{equation} | |
where $f_{o} : = T^{-1}\int_{o}^{T} f (t) dt$. For instance, | |
\begin{equation} | |
f (t) = \sum_{k\in \bbbn} \delta_{k} \xi\ , | |
\end{equation} | |
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: | |
\begin{equation} | |
H (t,x) = \frac{1}{2} \left(A_{\infty} (t) x,x\right) + N(t,x) | |
\end{equation} | |
\begin{equation} | |
\forall t\ ,\ \ \ N(t,x)\ \ \ \ \ | |
{\rm is\ convex\ with\ respect\ to}\ \ x | |
\end{equation} | |
\begin{equation} | |
N(t,x) \ge n\left(\left\|x\right\|\right)\ \ \ \ \ | |
{\rm with}\ \ n(s)s^{-1}\to +\infty\ \ {\rm as}\ \ s\to +\infty | |
\end{equation} | |
\begin{equation} | |
\exists c\in \bbbr\ :\ \ \ H (t,x) \le | |
\frac{1}{2} \left(B_{\infty} (t) x,x\right) + c\ \ \ \forall x\ . | |
\end{equation} | |
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 | |
\begin{equation} | |
H (t,x) = \frac{1}{2} k \left\|k\right\|^{2} +\left\|x\right\|^{\alpha} | |
\end{equation} | |
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{2rab}, 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{2clar:eke:2} to treat the | |
same problem in the convex-subquadratic case, with growth conditions on | |
$H$ only. | |
Recently, Michalek and Tarantello (see Michalek, R., Tarantello, G. | |
\cite{2mich:tar} and Tarantello, G. \cite{2tar}) 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}{} | |
% | |
\bibitem[1980]{2clar:eke} | |
Clarke, F., Ekeland, I.: | |
Nonlinear oscillations and | |
boundary-value problems for Hamiltonian systems. | |
Arch. Rat. Mech. Anal. 78, 315--333 (1982) | |
\bibitem[1981]{2clar:eke:2} | |
Clarke, F., Ekeland, I.: | |
Solutions p\'{e}riodiques, du | |
p\'{e}riode donn\'{e}e, des \'{e}quations hamiltoniennes. | |
Note CRAS Paris 287, 1013--1015 (1978) | |
\bibitem[1982]{2mich:tar} | |
Michalek, R., Tarantello, G.: | |
Subharmonic solutions with prescribed minimal | |
period for nonautonomous Hamiltonian systems. | |
J. Diff. Eq. 72, 28--55 (1988) | |
\bibitem[1983]{2tar} | |
Tarantello, G.: | |
Subharmonic solutions for Hamiltonian | |
systems via a $\bbbz_{p}$ pseudoindex theory. | |
Annali di Matematica Pura (to appear) | |
\bibitem[1985]{2rab} | |
Rabinowitz, P.: | |
On subharmonic solutions of a Hamiltonian system. | |
Comm. Pure Appl. Math. 33, 609--633 (1980) | |
\end{thebibliography} | |
\clearpage | |
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\addtocmark[2]{Subject Index} % additional numbered TOC entry | |
\markboth{Subject Index}{Subject Index} | |
\renewcommand{\indexname}{Subject Index} | |
\input{subjidx.tex} | |
\end{document} |