introduction multimedia
[]
readme
enquete
preface
I
1
2
II
3
4
III
5
6
7
IV
8
9
10
afterthoughts
appendix
references
resources
\documentclass{llncs}
%
\usepackage{makeidx} % allows for indexgeneration
%
\begin{document}
%
\frontmatter % for the preliminaries
%
\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.
\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)
%
\tocauthor{Ivar Ekeland (Princeton University),
Roger Temam (Universit\'{e} de Paris-Sud),
Jeffrey Dean, David Grove, Craig Chambers (Universit\`a di Geova),
Kim B. Bruce (Stanford University),
Elisa Bertino (Digita Research Center)}
%
\institute{Princeton University, Princeton NJ 08544, USA,\\
\email{I.Ekeland@princeton.edu},\\ WWW home page:
\texttt{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
\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 a convex function of x, going to when
%
\subsection{Autonomous Systems}
%
In this section, we will consider the case when the Hamiltonian
is autonomous. For the sake of simplicity, we shall also assume that it
is .
We shall first consider the question of nontriviality, within the
general framework of
-subquadratic Hamiltonians. In
the second subsection, we shall look into the special case when H is
-subquadratic,
and we shall try to derive additional information.
%
\subsubsection{The General Case: Nontriviality.}
%
We assume that H is
-sub\-qua\-dra\-tic at infinity,
for some constant symmetric matrices and ,
with 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}
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
, 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 , which is a subspace
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 and . Set:
\begin{equation}
\delta := \liminf_{x\to 0} 2 N (x) \left\|x\right\|^{-2}\ .
\end{equation}
If ,
the solution 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
, there is some 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 such that
\begin{equation}
f\left\|x\right\| \le \eta
\Rightarrow N^{\ast} (y) \le \frac{1}{2\delta '}
\left\|y\right\|^{2}\ .
\end{equation}
\begin{figure}
\caption{This is the caption of the figure displaying a white eagle and
a white horse on a snow field}
\end{figure}
Since is a smooth function, we will have
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 close enough to , the quantity
will be negative, and we end up with
\begin{equation}
\psi (hu_{1}) < 0\ \ \ \ \ for\ \ h\ne 0\ \ small\ .
\end{equation}
On the other hand, we check directly that . This shows
that 0 cannot be a minimizer of , not even a local one.
So and
. \qed
\end{proof}
%
\begin{corollary}
Assume H is and
-subquadratic at infinity. Let
be the
equilibria, that is, the solutions of .
Denote by
the smallest eigenvalue of , 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
\end{equation}
then minimization of yields a non-constant T-periodic solution
.
\end{corollary}
%
We recall once more that by the integer part of
, we mean the
such that . For instance,
if we take , Corollary 2 tells
us that 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)\ .
\end{equation}
%
\begin{proof}
The spectrum of is . The
largest negative eigenvalue is given by
,
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 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 on and
that is non-de\-gen\-er\-ate for any . Then any local
minimizer of has minimal period T.
\end{lemma}
%
\begin{proof}
We know that , or
for some constant , 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 . So
for all in some neighbourhood of x in
.
But this index is precisely the index
of the T-periodic
solution over the interval
$(0,T)\widetilde{x}x_{T}T\in \left(2\pi\omega^{-1}, 2\pi b_{\infty}^{-1}\right)x_{T} (0) = x_{T} (T)x_{T}T\to 2\pi \omega^{-1}H(t,x)-subquadratic at
infinity for all , and T-periodic in t
\begin{equation}
H (t,\cdot )\ \ \ \ \ {\rm is\ convex}\ \ \forall t
\end{equation}
\begin{equation}
H (\cdot ,x)\ \ \ \ \ is\ \ T-periodic\ \ \forall x
\end{equation}
\begin{equation}
H (t,x)\ge n\left(\left\|x\right\|\right)\ \ \ \ \
with\ \ n (s)s^{-1}\to \infty\ \ 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 , and is positive definite
everywhere. Then there is a sequence , , of
kT-periodic solutions of the system
\begin{equation}
\dot{x} = JH' (t,x)
\end{equation}
such that, for every , there is some 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
-subquadratic, and the
forcing term is a distribution on the circle:
\begin{equation}
f = \frac{d}{dt} F + f_{o}\ \ \ \ \
with\ \ F\in L^{2} \left(\bbbr / T\bbbz; \bbbr^{2n}\right)\ ,
\end{equation}
where . For instance,
\begin{equation}
f (t) = \sum_{k\in \bbbn} \delta_{k} \xi\ ,
\end{equation}
where is the Dirac mass at and
is a
constant, fits the prescription. This means that the system
is being excited by a
series of identical shocks at interval T.
\end{example}
%
\begin{definition}
Let and be symmetric
operators in , depending continuously on
, such that
for all t.
A Borelian function
is called
-subquadratic at infinity
if there exists a function 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)\ \ \ \ \
with\ \ n(s)s^{-1}\to +\infty\ \ 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 and
, with
,
we shall say that H is
at infinity. As an example, the function
$1\le \alpha < 2-subquadratic at infinity
for every . 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 )\varepsilon > 0k<0H'\bbbz_{p}