Hi Juan Ramon,
Please find attached the demo as I imported it into LyX using the ejour2 class
file. This is not appropriate, but it contains many of the things you need for
this document class.
The more or less successfully imported file is attached. However, note that I
broke a lot of things in doing so.
The proper thing to do is to add this document class to LyX. That is quite a
bit of work, involving the following:
1. Place the file llncs.cls in the directory where also article.cls, book.cls
etc. are. Then, run (as root) texhash, to make TeX find the file.
2. Create a file llncs.layout in the same directory that contains all the
other .layout files. You can copy ejour2.layout to llncs.layout, and edit it.
First thing to do is to edit the second line to read
# \DeclareLaTeXClass{Springer - Lecture Notes in Comp. Sci.}
Editing the llncs.layout file to contain all the commands and environments
present in the llncs class, is a big job. Study the example files. Once this
work is done, everybody will be able to use LyX for this class of document.
3. Then run configure from within LyX, exit and start again. You should now
have the llncs document class in layout->document menu.
Good luck,
Martin
#This file was created by <mv> Mon Jun 7 22:11:37 1999
#LyX 1.0 (C) 1995-1999 Matthias Ettrich and the LyX Team
\lyxformat 2.15
\textclass ejour2
\begin_preamble
%
\usepackage{makeidx} % allows for indexgeneration
%
\usepackage{multicol}
\def\bbbr{{\rm I\!R}} %reelle Zahlen
\def\bbbz{{\mathchoice {\hbox{$\mathsf\textstyle Z\kern-0.4em Z$}}
\@onecolumn
\end_preamble
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\papersize Default
\paperpackage a4
\use_geometry 0
\use_amsmath 0
\paperorientation portrait
\secnumdepth 3
\tocdepth 3
\paragraph_separation indent
\defskip medskip
\quotes_language english
\quotes_times 2
\papercolumns 1
\papersides 1
\paperpagestyle default
\layout Standard
\layout Standard
\latex latex
%
\backslash
frontmatter
\layout Standard
\latex latex
\backslash
pagestyle{headings}
\layout Standard
\latex latex
%
\backslash
addtocmark{Hamiltonian Mechanics}
\layout Standard
Preface
\layout Standard
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
\shape italic
The Physics of Atoms and Quanta
\shape default
.
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.
\layout Standard
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.
\layout Standard
\latex latex
\backslash
vspace{1cm}
\layout Standard
\noindent \align right
April 1995
\hfill
Walter Olthoff
\newline
Program Chair
\newline
ECOOP'95
\layout Standard
Organization
\layout Standard
ECOOP'95 is organized by the department of Computer Science, Univeristy
of
\latex latex
\backslash
AA
\latex default
rhus and AITO (association Internationa pour les Technologie Object) in
cooperation with ACM/SIGPLAN.
\layout Section*
Executive Commitee
\layout Standard
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Conference Chair:
\newline
Ole Lehrmann Madsen (
\latex latex
\backslash
AA
\latex default
rhus University, DK)
\newline
Program Chair:
\newline
Walter Olthoff (DFKI GmbH, Germany)
\newline
Organizing Chair:
\newline
J
\latex latex
\backslash
o
\latex default
rgen Lindskov Knudsen (
\latex latex
\backslash
AA
\latex default
rhus University, DK)
\newline
Tutorials:
\newline
Birger M
\latex latex
\backslash
o
\latex default
ller-Pedersen
\latex latex
\backslash
hfil
\backslash
break
\newline
\latex default
(Norwegian Computing Center, Norway)
\newline
Workshops:
\newline
Eric Jul (University of Kopenhagen, Denmark)
\newline
Panels:
\newline
Boris Magnusson (Lund University, Sweden)
\newline
Exhibition:
\newline
Elmer Sandvad (
\latex latex
\backslash
AA
\latex default
rhus University, DK)
\newline
Demonstrations:
\newline
Kurt N
\latex latex
\backslash
o
\latex default
rdmark (
\latex latex
\backslash
AA
\latex default
rhus University, DK)
\layout Section*
Program Commitee
\layout Standard
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Conference Chair:
\newline
Ole Lehrmann Madsen (
\latex latex
\backslash
AA
\latex default
rhus University, DK)
\newline
Program Chair:
\newline
Walter Olthoff (DFKI GmbH, Germany)
\newline
Organizing Chair:
\newline
J
\latex latex
\backslash
o
\latex default
rgen Lindskov Knudsen (
\latex latex
\backslash
AA
\latex default
rhus University, DK)
\newline
Tutorials:
\newline
Birger M
\latex latex
\backslash
o
\latex default
ller-Pedersen
\latex latex
\backslash
hfil
\backslash
break
\newline
\latex default
(Norwegian Computing Center, Norway)
\newline
Workshops:
\newline
Eric Jul (University of Kopenhagen, Denmark)
\newline
Panels:
\newline
Boris Magnusson (Lund University, Sweden)
\newline
Exhibition:
\newline
Elmer Sandvad (
\latex latex
\backslash
AA
\latex default
rhus University, DK)
\newline
Demonstrations:
\newline
Kurt N
\latex latex
\backslash
o
\latex default
rdmark (
\latex latex
\backslash
AA
\latex default
rhus University, DK)
\layout Standard
\latex latex
\backslash
begin{multicols}{3}[
\backslash
section*{Referees}]
\newline
V.~Andreev
\backslash
\backslash
\newline
B
\backslash
"{a}rwolff
\backslash
\backslash
\newline
E.~Barrelet
\backslash
\backslash
\newline
H.P.~Beck
\backslash
\backslash
\newline
G.~Bernardi
\backslash
\backslash
\newline
E.~Binder
\backslash
\backslash
\newline
P.C.~Bosetti
\backslash
\backslash
\newline
Braunschweig
\backslash
\backslash
\newline
F.W.~B
\backslash
"{u}sser
\backslash
\backslash
\newline
T.~Carli
\backslash
\backslash
\newline
A.B.~Clegg
\backslash
\backslash
\newline
G.~Cozzika
\backslash
\backslash
\newline
S.~Dagoret
\backslash
\backslash
\newline
Del~Buono
\backslash
\backslash
\newline
P.~Dingus
\backslash
\backslash
\newline
H.~Duhm
\backslash
\backslash
\newline
J.~Ebert
\backslash
\backslash
\newline
S.~Eichenberger
\backslash
\backslash
\newline
R.J.~Ellison
\backslash
\backslash
\newline
Feltesse
\backslash
\backslash
\newline
W.~Flauger
\backslash
\backslash
\newline
A.~Fomenko
\backslash
\backslash
\newline
G.~Franke
\backslash
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\newline
J.~Garvey
\backslash
\backslash
\newline
M.~Gennis
\backslash
\backslash
\newline
L.~Goerlich
\backslash
\backslash
\newline
P.~Goritchev
\backslash
\backslash
\newline
H.~Greif
\backslash
\backslash
\newline
E.M.~Hanlon
\backslash
\backslash
\newline
R.~Haydar
\backslash
\backslash
\newline
R.C.W.~Henderso
\backslash
\backslash
\newline
P.~Hill
\backslash
\backslash
\newline
H.~Hufnagel
\backslash
\backslash
\newline
A.~Jacholkowska
\backslash
\backslash
\newline
Johannsen
\backslash
\backslash
\newline
S.~Kasarian
\backslash
\backslash
\newline
I.R.~Kenyon
\backslash
\backslash
\newline
C.~Kleinwort
\backslash
\backslash
\newline
T.~K
\backslash
"{o}hler
\backslash
\backslash
\newline
S.D.~Kolya
\backslash
\backslash
\newline
P.~Kostka
\backslash
\backslash
\newline
U.~Kr
\backslash
"{u}ger
\backslash
\backslash
\newline
J.~Kurzh
\backslash
"{o}fer
\backslash
\backslash
\newline
M.P.J.~Landon
\backslash
\backslash
\newline
A.~Lebedev
\backslash
\backslash
\newline
Ch.~Ley
\backslash
\backslash
\newline
F.~Linsel
\backslash
\backslash
\newline
H.~Lohmand
\backslash
\backslash
\newline
Martin
\backslash
\backslash
\newline
S.~Masson
\backslash
\backslash
\newline
K.~Meier
\backslash
\backslash
\newline
C.A.~Meyer
\backslash
\backslash
\newline
S.~Mikocki
\backslash
\backslash
\newline
J.V.~Morris
\backslash
\backslash
\newline
B.~Naroska
\backslash
\backslash
\newline
Nguyen
\backslash
\backslash
\newline
U.~Obrock
\backslash
\backslash
\newline
G.D.~Patel
\backslash
\backslash
\newline
Ch.~Pichler
\backslash
\backslash
\newline
S.~Prell
\backslash
\backslash
\newline
F.~Raupach
\backslash
\backslash
\newline
V.~Riech
\backslash
\backslash
\newline
P.~Robmann
\backslash
\backslash
\newline
N.~Sahlmann
\backslash
\backslash
\newline
P.~Schleper
\backslash
\backslash
\newline
Sch
\backslash
"{o}ning
\backslash
\backslash
\newline
B.~Schwab
\backslash
\backslash
\newline
A.~Semenov
\backslash
\backslash
\newline
G.~Siegmon
\backslash
\backslash
\newline
J.R.~Smith
\backslash
\backslash
\newline
M.~Steenbock
\backslash
\backslash
\newline
U.~Straumann
\backslash
\backslash
\newline
C.~Thiebaux
\backslash
\backslash
\newline
P.~Van~Esch
\backslash
\backslash
\newline
from Yerevan Ph
\backslash
\backslash
\newline
L.R.~West
\backslash
\backslash
\newline
G.-G.~Winter
\backslash
\backslash
\newline
T.P.~Yiou
\backslash
\backslash
\newline
M.~Zimmer
\backslash
end{multicols}
\layout Section*
Sponsoring Institutions
\layout Standard
Bernauer-Budiman Inc., Reading, Mass.
\newline
The Hofmann-International Company, San Louis Obispo, Cal.
\newline
Kramer Industries, Heidelberg, Germany
\begin_inset LatexCommand \tableofcontents{}
\end_inset
\latex latex
%
\backslash
mainmatter
\layout Title
Hamiltonian Mechanics unter besonderer Ber\i \"{u}
cksichtigung der h\i \"{o}
hreren Lehranstalten
\layout Running LaTeX Title
Hamiltonian Mechanics
\layout Author
Ivar Ekeland
\latex latex
\backslash
inst{1}
\backslash
and
\latex default
Roger Temam
\latex latex
\backslash
inst{2}
\latex default
Jeffrey Dean
\latex latex
\backslash
and
\latex default
David Grove
\latex latex
\backslash
and
\latex default
Craig Chambers
\latex latex
\backslash
and
\latex default
Kim
\protected_separator
B.
\protected_separator
Bruce
\latex latex
\backslash
and
\newline
\latex default
Elsa Bertino
\layout Author Running
Ivar Ekeland et al.
\layout Standard
\latex latex
%
\backslash
tocauthor{Ivar Ekeland (Universit
\backslash
'{e} de Paris-Sud),
\newline
Jeffrey Dean, David Grove, Craig Chambers (Universit
\backslash
`{a} di Geova),
\newline
Kim B.
Bruce (Stanford University),
\newline
%Elisa Bertino (Digita Research Center)}
\layout Institute
Princeton University, Princeton NJ 08544, USA,
\newline
\latex latex
\backslash
email{[EMAIL PROTECTED]}
\latex default
,
\newline
WWW home page:
\family typewriter
http://users/
\latex latex
\backslash
homedir
\latex default
iekeland/web/welcome.html
\family default
\latex latex
\backslash
and
\newline
\latex default
Universit\i \'{e}
de Paris-Sud, Laboratoire d'Analyse Num\i \'{e}
rique, B\i \^{a}
timent 425,
\newline
F-91405 Orsay Cedex, France
\layout 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.
\latex latex
\backslash
dots
\newline
\layout Section
Fixed-Period Problems: The Sublinear Case
\layout Standard
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_inset Formula
\begin{eqnarray*}
\dot{x} & = & JH'(t,x)\\
x(0) & = & x(T)
\end{eqnarray*}
\end_inset
with
\begin_inset Formula \( H(t,\cdot ) \)
\end_inset
a convex function of
\begin_inset Formula \( x \)
\end_inset
, going to
\begin_inset Formula \( +\infty \)
\end_inset
when
\begin_inset Formula \( \left\Vert x\right\Vert \rightarrow \infty \)
\end_inset
.
\layout Subsection
Autonomous Systems
\layout Standard
In this section, we will consider the case when the Hamiltonian
\begin_inset Formula \( H(x) \)
\end_inset
is autonomous.
For the sake of simplicity, we shall also assume that it is
\begin_inset Formula \( C^{1} \)
\end_inset
.
\layout Standard
We shall first consider the question of nontriviality, within the general
framework of
\begin_inset Formula \( \left( A_{\infty },B_{\infty }\right) \)
\end_inset
-subquadratic Hamiltonians.
In the second subsection, we shall look into the special case when
\begin_inset Formula \( H \)
\end_inset
is
\begin_inset Formula \( \left( 0,b_{\infty }\right) \)
\end_inset
-subquadratic, and we shall try to derive additional information.
\layout Subsubsection
The General Case: Nontriviality.
\layout Standard
We assume that
\begin_inset Formula \( H \)
\end_inset
is
\begin_inset Formula \( \left( A_{\infty },B_{\infty }\right) \)
\end_inset
-sub\SpecialChar \-
qua\SpecialChar \-
dra\SpecialChar \-
tic at infinity, for some constant symmetric matrices
\begin_inset Formula \( A_{\infty } \)
\end_inset
and
\begin_inset Formula \( B_{\infty } \)
\end_inset
, with
\begin_inset Formula \( B_{\infty }-A_{\infty } \)
\end_inset
positive definite.
Set:
\begin_inset Formula
\begin{eqnarray}
\gamma : & = & {\textrm{smallest}\, \, \textrm{eigenvalue}\, \, \textrm{of}}\, \, \,
\, B_{\infty }-A_{\infty }\\
\lambda : & = & {\textrm{largest}\, \, \textrm{negative}\, \, \textrm{eigenvalue}\, \,
\textrm{of}}\, \, J\frac{d}{dt}+A_{\infty }\, \, .
\end{eqnarray}
\end_inset
\layout Standard
Theorem
\protected_separator
\begin_inset LatexCommand \ref{ghou:pre}
\end_inset
tells us that if
\begin_inset Formula \( \lambda +\gamma <0 \)
\end_inset
, the boundary-value problem:
\begin_inset Formula
\begin{equation}
\begin{array}{rcl}
\dot{x} & = & JH'(x)\\
x(0) & = & x(T)
\end{array}
\end{equation}
\end_inset
has at least one solution
\begin_inset Formula \( \overline{x} \)
\end_inset
, which is found by minimizing the dual action functional:
\begin_inset Formula
\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}
\end_inset
on the range of
\begin_inset Formula \( \Lambda \)
\end_inset
, which is a subspace
\begin_inset Formula \( R(\Lambda )_{L}^{2} \)
\end_inset
with finite codimension.
Here
\begin_inset Formula
\begin{equation}
N(x):=H(x)-\frac{1}{2}\left( A_{\infty }x,x\right)
\end{equation}
\end_inset
is a convex function, and
\begin_inset Formula
\begin{equation}
N(x)\leq \frac{1}{2}\left( \left( B_{\infty }-A_{\infty }\right) x,x\right) +c\, \, \,
\, \, \, \forall x\, \, .
\end{equation}
\end_inset
\layout Proposition
Assume
\begin_inset Formula \( H'(0)=0 \)
\end_inset
and
\begin_inset Formula \( H(0)=0 \)
\end_inset
.
Set:
\begin_inset Formula
\begin{equation}
\label{eq:one}
\delta :=\liminf _{x\rightarrow 0}2N(x)\left\Vert x\right\Vert ^{-2}\, \, .
\end{equation}
\end_inset
\layout Proposition
If
\begin_inset Formula \( \gamma <-\lambda <\delta \)
\end_inset
, the solution
\begin_inset Formula \( \overline{u} \)
\end_inset
is non-zero:
\begin_inset Formula
\begin{equation}
\overline{x}(t)\ne 0\, \, \, \, \, \, \forall t\, \, .
\end{equation}
\end_inset
\layout Proof
Condition (
\begin_inset LatexCommand \ref{eq:one}
\end_inset
) means that, for every
\begin_inset Formula \( \delta '>\delta \)
\end_inset
, there is some
\begin_inset Formula \( \varepsilon >0 \)
\end_inset
such that
\begin_inset Formula
\begin{equation}
\left\Vert x\right\Vert \leq \varepsilon \Rightarrow N(x)\leq \frac{\delta
'}{2}\left\Vert x\right\Vert ^{2}\, \, .
\end{equation}
\end_inset
\layout Proof
It is an exercise in convex analysis, into which we shall not go, to show
that this implies that there is an
\begin_inset Formula \( \eta >0 \)
\end_inset
such that
\begin_inset Formula
\begin{equation}
\label{eq:two}
f\left\Vert x\right\Vert \leq \eta \Rightarrow N^{\ast }(y)\leq \frac{1}{2\delta
'}\left\Vert y\right\Vert ^{2}\, \, .
\end{equation}
\end_inset
\begin_float fig
\layout Proof
\latex latex
\backslash
vspace{2.5cm}
\begin_deeper
\layout Caption
This is the caption of the figure displaying a white eagle and a white horse
on a snow field
\end_float
\end_deeper
\layout Proof
Since
\begin_inset Formula \( u_{1} \)
\end_inset
is a smooth function, we will have
\begin_inset Formula \( \left\Vert hu_{1}\right\Vert _{\infty }\leq \eta \)
\end_inset
for
\begin_inset Formula \( h \)
\end_inset
small enough, and inequality (
\begin_inset LatexCommand \ref{eq:two}
\end_inset
) will hold, yielding thereby:
\begin_inset Formula
\begin{equation}
\psi (hu_{1})\leq \frac{h^{2}}{2}\frac{1}{\lambda }\left\Vert u_{1}\right\Vert
_{2}^{2}+\frac{h^{2}}{2}\frac{1}{\delta '}\left\Vert u_{1}\right\Vert ^{2}\, \, .
\end{equation}
\end_inset
\layout Proof
If we choose
\begin_inset Formula \( \delta ' \)
\end_inset
close enough to
\begin_inset Formula \( \delta \)
\end_inset
, the quantity
\begin_inset Formula \( \left( \frac{1}{\lambda }+\frac{1}{\delta '}\right) \)
\end_inset
will be negative, and we end up with
\begin_inset Formula
\begin{equation}
\psi (hu_{1})<0\, \, \, \, \, \, \, \, \, \, {\textrm{for}}\, \, \, \, h\ne 0\, \, \,
\, {\textrm{small}}\, \, .
\end{equation}
\end_inset
\layout Proof
On the other hand, we check directly that
\begin_inset Formula \( \psi (0)=0 \)
\end_inset
.
This shows that 0 cannot be a minimizer of
\begin_inset Formula \( \psi \)
\end_inset
, not even a local one.
So
\begin_inset Formula \( \overline{u}\ne 0 \)
\end_inset
and
\begin_inset Formula \( \overline{u}\ne \Lambda _{o}^{-1}(0)=0 \)
\end_inset
.
\latex latex
\backslash
qed
\newline
\layout Corollary
Assume
\begin_inset Formula \( H \)
\end_inset
is
\begin_inset Formula \( C^{2} \)
\end_inset
and
\begin_inset Formula \( \left( a_{\infty },b_{\infty }\right) \)
\end_inset
-subquadratic at infinity.
Let
\begin_inset Formula \( \xi _{1},\allowbreak \dots ,\allowbreak \xi _{N} \)
\end_inset
be the equilibria, that is, the solutions of
\begin_inset Formula \( H'(\xi )=0 \)
\end_inset
.
Denote by
\begin_inset Formula \( \omega _{k} \)
\end_inset
the smallest eigenvalue of
\begin_inset Formula \( H''\left( \xi _{k}\right) \)
\end_inset
, and set:
\begin_inset Formula
\begin{equation}
\omega :={\textrm{Min}\, }\left\{ \omega _{1},\dots ,\omega _{k}\right\} \, \, .
\end{equation}
\end_inset
If:
\begin_inset Formula
\begin{equation}
\label{eq:three}
\frac{T}{2\pi }b_{\infty }<-E\left[ -\frac{T}{2\pi }a_{\infty }\right] <\frac{T}{2\pi
}\omega
\end{equation}
\end_inset
then minimization of
\begin_inset Formula \( \psi \)
\end_inset
yields a non-constant
\begin_inset Formula \( T \)
\end_inset
-periodic solution
\begin_inset Formula \( \overline{x} \)
\end_inset
.
\layout Standard
We recall once more that by the integer part
\begin_inset Formula \( E[\alpha ] \)
\end_inset
of
\begin_inset Formula \( \alpha \in \bbbr \)
\end_inset
, we mean the
\begin_inset Formula \( a\in \bbbz \)
\end_inset
such that
\begin_inset Formula \( a<\alpha \leq a+1 \)
\end_inset
.
For instance, if we take
\begin_inset Formula \( a_{\infty }=0 \)
\end_inset
, Corollary 2 tells us that
\begin_inset Formula \( \overline{x} \)
\end_inset
exists and is non-constant provided that:
\layout Standard
\begin_inset Formula
\begin{equation}
\frac{T}{2\pi }b_{\infty }<1<\frac{T}{2\pi }
\end{equation}
\end_inset
or
\begin_inset Formula
\begin{equation}
\label{eq:four}
T\in \left( \frac{2\pi }{\omega },\frac{2\pi }{b_{\infty }}\right) \, \, .
\end{equation}
\end_inset
\layout Proof
The spectrum of
\begin_inset Formula \( \Lambda \)
\end_inset
is
\begin_inset Formula \( \frac{2\pi }{T}\bbbz +a_{\infty } \)
\end_inset
.
The largest negative eigenvalue
\begin_inset Formula \( \lambda \)
\end_inset
is given by
\begin_inset Formula \( \frac{2\pi }{T}k_{o}+a_{\infty } \)
\end_inset
, where
\begin_inset Formula
\begin{equation}
\frac{2\pi }{T}k_{o}+a_{\infty }<0\leq \frac{2\pi }{T}(k_{o}+1)+a_{\infty }\, \, .
\end{equation}
\end_inset
Hence:
\begin_inset Formula
\begin{equation}
k_{o}=E\left[ -\frac{T}{2\pi }a_{\infty }\right] \, \, .
\end{equation}
\end_inset
\layout Proof
The condition
\begin_inset Formula \( \gamma <-\lambda <\delta \)
\end_inset
now becomes:
\begin_inset Formula
\begin{equation}
b_{\infty }-a_{\infty }<-\frac{2\pi }{T}k_{o}-a_{\infty }<\omega -a_{\infty }
\end{equation}
\end_inset
which is precisely condition (
\begin_inset LatexCommand \ref{eq:three}
\end_inset
).
\latex latex
\backslash
qed
\newline
\layout Lemma
Assume that
\begin_inset Formula \( H \)
\end_inset
is
\begin_inset Formula \( C^{2} \)
\end_inset
on
\begin_inset Formula \( \bbbr ^{2n}\setminus \{0\} \)
\end_inset
and that
\begin_inset Formula \( H''(x) \)
\end_inset
is non-de\SpecialChar \-
gen\SpecialChar \-
er\SpecialChar \-
ate for any
\begin_inset Formula \( x\ne 0 \)
\end_inset
.
Then any local minimizer
\begin_inset Formula \( \widetilde{x} \)
\end_inset
of
\begin_inset Formula \( \psi \)
\end_inset
has minimal period
\begin_inset Formula \( T \)
\end_inset
.
\layout Proof
We know that
\begin_inset Formula \( \widetilde{x} \)
\end_inset
, or
\begin_inset Formula \( \widetilde{x}+\xi \)
\end_inset
for some constant
\begin_inset Formula \( \xi \in \bbbr ^{2n} \)
\end_inset
, is a
\begin_inset Formula \( T \)
\end_inset
-periodic solution of the Hamiltonian system:
\begin_inset Formula
\begin{equation}
\dot{x}=JH'(x)\, \, .
\end{equation}
\end_inset
\layout Proof
There is no loss of generality in taking
\begin_inset Formula \( \xi =0 \)
\end_inset
.
So
\begin_inset Formula \( \psi (x)\geq \psi (\widetilde{x}) \)
\end_inset
for all
\begin_inset Formula \( \widetilde{x} \)
\end_inset
in some neighbourhood of
\begin_inset Formula \( x \)
\end_inset
in
\begin_inset Formula \( W^{1,2}\left( \bbbr /T\bbbz ;\bbbr ^{2n}\right) \)
\end_inset
.
\layout Proof
But this index is precisely the index
\begin_inset Formula \( i_{T}(\widetilde{x}) \)
\end_inset
of the
\begin_inset Formula \( T \)
\end_inset
-periodic solution
\begin_inset Formula \( \widetilde{x} \)
\end_inset
over the interval
\begin_inset Formula \( (0,T) \)
\end_inset
, as defined in Sect.
\protected_separator
2.6.
So
\begin_inset Formula
\begin{equation}
\label{eq:five}
i_{T}(\widetilde{x})=0\, \, .
\end{equation}
\end_inset
\layout Proof
Now if
\begin_inset Formula \( \widetilde{x} \)
\end_inset
has a lower period,
\begin_inset Formula \( T/k \)
\end_inset
say, we would have, by Corollary 31:
\begin_inset Formula
\begin{equation}
i_{T}(\widetilde{x})=i_{kT/k}(\widetilde{x})\geq ki_{T/k}(\widetilde{x})+k-1\geq
k-1\geq 1\, \, .
\end{equation}
\end_inset
\layout Proof
This would contradict (
\begin_inset LatexCommand \ref{eq:five}
\end_inset
), and thus cannot happen.
\latex latex
\backslash
qed
\newline
\layout Paragraph
Notes and Comments.
\layout Standard
The results in this section are a refined version of
\begin_inset LatexCommand \cite{clar:eke}
\end_inset
; the minimality result of Proposition 14 was the first of its kind.
\layout Standard
To understand the nontriviality conditions, such as the one in formula (
\begin_inset LatexCommand \ref{eq:four}
\end_inset
), one may think of a one-parameter family
\begin_inset Formula \( x_{T} \)
\end_inset
,
\begin_inset Formula \( T\in \left( 2\pi \omega ^{-1},2\pi b_{\infty }^{-1}\right) \)
\end_inset
of periodic solutions,
\begin_inset Formula \( x_{T}(0)=x_{T}(T) \)
\end_inset
, with
\begin_inset Formula \( x_{T} \)
\end_inset
going away to infinity when
\begin_inset Formula \( T\rightarrow 2\pi \omega ^{-1} \)
\end_inset
, which is the period of the linearized system at 0.
\begin_float tab
\layout Caption
This is the example table taken out of
\shape italic
The TeXbook,
\shape default
p.
\latex latex
\backslash
,
\latex default
246
\layout Standard
\align center \LyXTable
multicol5
6 3 0 0 0 0 0 0
1 1 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 1 0 0
8 0 0 "" "r@{\quad}"
4 0 0 "" ""
2 0 0 "" ""
1 2 1 1 0 0 0 "" ""
1 2 1 1 0 0 0 "" ""
2 2 1 1 0 0 0 "" ""
0 8 0 0 0 0 0 "" ""
0 4 0 0 0 0 0 "" ""
0 2 0 0 0 0 0 "" ""
0 8 0 0 0 0 0 "" ""
0 4 0 0 0 0 0 "" ""
0 2 0 0 0 0 0 "" ""
0 8 0 0 0 0 0 "" ""
0 4 0 0 0 0 0 "" ""
0 2 0 0 0 0 0 "" ""
0 8 0 0 0 0 0 "" ""
0 4 0 0 0 0 0 "" ""
0 2 0 0 0 0 0 "" ""
0 8 0 1 0 0 0 "" ""
0 4 0 1 0 0 0 "" ""
0 2 0 1 0 0 0 "" ""
\latex latex
\backslash
rule{0pt}{12pt}
\latex default
Year
\newline
World population
\newline
\latex latex
\backslash
rule{0pt}{12pt}
\latex default
8000 B.C.
\newline
5,000,000
\newline
\newline
50 A.D.
\newline
200,000,000
\newline
\newline
1650 A.D.
\newline
500,000,000
\newline
\newline
1945 A.D.
\newline
2,300,000,000
\newline
\newline
1980 A.D.
\newline
4,400,000,000
\newline
\end_float
\layout Theorem
[Ghoussoub-Preiss]
\begin_inset LatexCommand \label{ghou:pre}
\end_inset
Assume
\begin_inset Formula \( H(t,x) \)
\end_inset
is
\begin_inset Formula \( (0,\varepsilon ) \)
\end_inset
-subquadratic at infinity for all
\begin_inset Formula \( \varepsilon >0 \)
\end_inset
, and
\begin_inset Formula \( T \)
\end_inset
-periodic in
\begin_inset Formula \( t \)
\end_inset
\begin_inset Formula
\begin{equation}
H(t,\cdot )\, \, \, \, \, \, \, \, \, \, {\textrm{is}\, \, \textrm{convex}}\, \, \, \,
\forall t
\end{equation}
\end_inset
\begin_inset Formula
\begin{equation}
H(\cdot ,x)\, \, \, \, \, \, \, \, \, \, {\textrm{is}}\, \, \, \,
T{-\textrm{periodic}}\, \, \, \, \forall x
\end{equation}
\end_inset
\begin_inset Formula
\begin{equation}
H(t,x)\geq n\left( \left\Vert x\right\Vert \right) \, \, \, \, \, \, \, \,
{\textrm{with}}\, \, \, \, n(s)s^{-1}\rightarrow \infty \, \, \, \, {\textrm{as}}\, \,
\, \, s\rightarrow \infty
\end{equation}
\end_inset
\begin_inset Formula
\begin{equation}
\forall \varepsilon >0\, \, ,\, \, \, \, \, \, \exists c\, \, :H(t,x)\leq
\frac{\varepsilon }{2}\left\Vert x\right\Vert ^{2}+c\, \, .
\end{equation}
\end_inset
\layout Theorem
Assume also that
\begin_inset Formula \( H \)
\end_inset
is
\begin_inset Formula \( C^{2} \)
\end_inset
, and
\begin_inset Formula \( H''(t,x) \)
\end_inset
is positive definite everywhere.
Then there is a sequence
\begin_inset Formula \( x_{k} \)
\end_inset
,
\begin_inset Formula \( k\in \bbbn \)
\end_inset
, of
\begin_inset Formula \( kT \)
\end_inset
-periodic solutions of the system
\begin_inset Formula
\begin{equation}
\dot{x}=JH'(t,x)
\end{equation}
\end_inset
such that, for every
\begin_inset Formula \( k\in \bbbn \)
\end_inset
, there is some
\begin_inset Formula \( p_{o}\in \bbbn \)
\end_inset
with:
\begin_inset Formula
\begin{equation}
p\geq p_{o}\Rightarrow x_{pk}\ne x_{k}\, \, .
\end{equation}
\end_inset
\latex latex
\backslash
qed
\newline
\layout Example
[
\family roman
External forcing
\family default
] Consider the system:
\begin_inset Formula
\begin{equation}
\dot{x}=JH'(x)+f(t)
\end{equation}
\end_inset
where the Hamiltonian
\begin_inset Formula \( H \)
\end_inset
is
\begin_inset Formula \( \left( 0,b_{\infty }\right) \)
\end_inset
-subquadratic, and the forcing term is a distribution on the circle:
\begin_inset Formula
\begin{equation}
f=\frac{d}{dt}F+f_{o}\, \, \, \, \, \, \, \, {\textrm{with}}\, \, \, \, F\in
L^{2}\left( \bbbr /T\bbbz ;\bbbr ^{2n}\right) \, \, ,
\end{equation}
\end_inset
where
\begin_inset Formula \( f_{o}:=T^{-1}\int _{o}^{T}f(t)dt \)
\end_inset
.
For instance,
\begin_inset Formula
\begin{equation}
f(t)=\sum _{k\in \bbbn }\delta _{k}\xi \, \, ,
\end{equation}
\end_inset
where
\begin_inset Formula \( \delta _{k} \)
\end_inset
is the Dirac mass at
\begin_inset Formula \( t=k \)
\end_inset
and
\begin_inset Formula \( \xi \in \bbbr ^{2n} \)
\end_inset
is a constant, fits the prescription.
This means that the system
\begin_inset Formula \( \dot{x}=JH'(x) \)
\end_inset
is being excited by a series of identical shocks at interval
\begin_inset Formula \( T \)
\end_inset
.
\layout Definition
Let
\begin_inset Formula \( A_{\infty }(t) \)
\end_inset
and
\begin_inset Formula \( B_{\infty }(t) \)
\end_inset
be symmetric operators in
\begin_inset Formula \( \bbbr ^{2n} \)
\end_inset
, depending continuously on
\begin_inset Formula \( t\in [0,T] \)
\end_inset
, such that
\begin_inset Formula \( A_{\infty }(t)\leq B_{\infty }(t) \)
\end_inset
for all
\begin_inset Formula \( t \)
\end_inset
.
\layout Definition
A Borelian function
\begin_inset Formula \( H:[0,T]\times \bbbr ^{2n}\rightarrow \bbbr \)
\end_inset
is called
\begin_inset Formula \( \left( A_{\infty },B_{\infty }\right) \)
\end_inset
-
\shape italic
subquadratic at infinity
\shape default
if there exists a function
\begin_inset Formula \( N(t,x) \)
\end_inset
such that:
\begin_inset Formula
\begin{equation}
H(t,x)=\frac{1}{2}\left( A_{\infty }(t)x,x\right) +N(t,x)
\end{equation}
\end_inset
\begin_inset Formula
\begin{equation}
\forall t\, \, ,\, \, \, \, \, \, N(t,x)\, \, \, \, \, \, \, \, {\textrm{is}\, \,
\textrm{convex}\, \, \textrm{with}\, \, \textrm{ respect}\, \, \textrm{ to}}\, \, \,
\, x
\end{equation}
\end_inset
\begin_inset Formula
\begin{equation}
N(t,x)\geq n\left( \left\Vert x\right\Vert \right) \, \, \, \, \, \, \, \,
{\textrm{with}}\, \, \, \, n(s)s^{-1}\rightarrow +\infty \, \, \, \, {\textrm{as}}\,
\, \, \, s\rightarrow +\infty
\end{equation}
\end_inset
\begin_inset Formula
\begin{equation}
\exists c\in \bbbr \, \, :\, \, \, \, \, \, H(t,x)\leq \frac{1}{2}\left( B_{\infty
}(t)x,x\right) +c\, \, \, \, \, \, \forall x\, \, .
\end{equation}
\end_inset
\layout Definition
If
\begin_inset Formula \( A_{\infty }(t)=a_{\infty }I \)
\end_inset
and
\begin_inset Formula \( B_{\infty }(t)=b_{\infty }I \)
\end_inset
, with
\begin_inset Formula \( a_{\infty }\leq b_{\infty }\in \bbbr \)
\end_inset
, we shall say that
\begin_inset Formula \( H \)
\end_inset
is
\begin_inset Formula \( \left( a_{\infty },b_{\infty }\right) \)
\end_inset
-subquadratic at infinity.
As an example, the function
\begin_inset Formula \( \left\Vert x\right\Vert ^{\alpha } \)
\end_inset
, with
\begin_inset Formula \( 1\leq \alpha <2 \)
\end_inset
, is
\begin_inset Formula \( (0,\varepsilon ) \)
\end_inset
-subquadratic at infinity for every
\begin_inset Formula \( \varepsilon >0 \)
\end_inset
.
Similarly, the Hamiltonian
\begin_inset Formula
\begin{equation}
H(t,x)=\frac{1}{2}k\left\Vert k\right\Vert ^{2}+\left\Vert x\right\Vert ^{\alpha }
\end{equation}
\end_inset
is
\begin_inset Formula \( (k,k+\varepsilon ) \)
\end_inset
-subquadratic for every
\begin_inset Formula \( \varepsilon >0 \)
\end_inset
.
Note that, if
\begin_inset Formula \( k<0 \)
\end_inset
, it is not convex.
\layout Paragraph
Notes and Comments.
\layout Standard
The first results on subharmonics were obtained by Rabinowitz in
\begin_inset LatexCommand \cite{rab}
\end_inset
, who showed the existence of infinitely many subharmonics both in the subquadra
tic and superquadratic case, with suitable growth conditions on
\begin_inset Formula \( H' \)
\end_inset
.
Again the duality approach enabled Clarke and Ekeland in
\begin_inset LatexCommand \cite{clar:eke:2}
\end_inset
to treat the same problem in the convex-subquadratic case, with growth
conditions on
\begin_inset Formula \( H \)
\end_inset
only.
\layout Standard
Recently, Michalek and Tarantello (see
\begin_inset LatexCommand \cite{mich:tar}
\end_inset
and
\begin_inset LatexCommand \cite{tar}
\end_inset
) have obtained lower bound on the number of subharmonics of period
\begin_inset Formula \( kT \)
\end_inset
, based on symmetry considerations and on pinching estimates, as in Sect.
\protected_separator
5.2 of this article.
\layout Standard
\latex latex
\backslash
begin{thebibliography}{5}
\newline
\backslash
bibitem {clar:eke}
\newline
Clarke, F., Ekeland, I.:
\newline
Nonlinear oscillations and
\newline
boundary-value problems for Hamiltonian systems.
\newline
Arch.
Rat.
Mech.
Anal.
{
\backslash
textbf{78}} (1982) 315--333
\newline
\newline
\backslash
bibitem {clar:eke:2}
\newline
Clarke, F., Ekeland, I.:
\newline
Solutions p
\backslash
'{e}riodiques, du
\newline
p
\backslash
'{e}riode donn
\backslash
'{e}e, des
\backslash
'{e}quations hamiltoniennes.
\newline
Note CRAS Paris {
\backslash
textbf{287}} (1978) 1013--1015
\newline
\newline
\backslash
bibitem {mich:tar}
\newline
Michalek, R., Tarantello, G.:
\newline
Subharmonic solutions with prescribed minimal
\newline
period for nonautonomous Hamiltonian systems.
\newline
J.
Diff.
Eq.
{
\backslash
textbf{72}} (1988) 28--55
\newline
\newline
\backslash
bibitem {tar}
\newline
Tarantello, G.:
\newline
Subharmonic solutions for Hamiltonian
\newline
systems via a
\backslash
(
\backslash
bbbz_{p}
\backslash
) pseudoindex theory.
\newline
Annali di Matematica Pura (to appear)
\newline
\newline
\backslash
bibitem {rab}
\newline
Rabinowitz, P.:
\newline
On subharmonic solutions of a Hamiltonian system.
\newline
Comm.
Pure Appl.
Math.
{
\backslash
textbf{33}} (1980) 609--633
\newline
\newline
\backslash
end{thebibliography}
\layout Title
Hamiltonian Mechanics2
\layout Author
Ivar Ekeland
\latex latex
\backslash
inst{1}
\backslash
and
\latex default
Roger Temam
\latex latex
\backslash
inst{2}
\layout Institute
Princeton University, Princeton NJ 08544, USA
\latex latex
\backslash
and
\newline
\latex default
Universit\i \'{e}
de Paris-Sud, Laboratoire d'Analyse Num\i \'{e}
rique, B\i \^{a}
timent 425,
\newline
F-91405 Orsay Cedex, France
\layout Standard
\latex latex
\backslash
makeatletter
\newline
\backslash
renewenvironment{thebibliography}[1]{
\newline
\backslash
section*{
\backslash
refname
\newline
\backslash
small
\newline
\backslash
list{}{
\backslash
settowidth{
\backslash
labelwidth}{}
\backslash
leftmargin
\backslash
parindent
\newline
\backslash
itemindent=-
\backslash
parindent
\newline
\backslash
labelsep=
\backslash
z@
\newline
\backslash
if@openbib
\newline
\backslash
advance
\backslash
leftmargin
\backslash
bibindent
\newline
\backslash
itemindent -
\backslash
bibindent
\newline
\backslash
listparindent
\backslash
itemindent
\newline
\backslash
parsep
\backslash
z@
\newline
\backslash
fi
\newline
\backslash
usecounter{enumiv}
\backslash
let
\backslash
p@enumiv
\backslash
@empty
\newline
\backslash
renewcommand{
\backslash
theenumiv}{}}
\backslash
if@openbib
\newline
\backslash
renewcommand{
\backslash
newblock}{
\backslash
par}
\backslash
else
\newline
\backslash
renewcommand{
\backslash
newblock}{
\backslash
hskip .11em
\backslash
@plus.33em
\backslash
@minus.07em}
\backslash
fi
\newline
\backslash
sloppy
\backslash
clubpenalty4000
\backslash
widowpenalty4000
\backslash
sfcode`
\backslash
.{=}
\backslash
@m}}{
\newline
{
\backslash
def
\backslash
@noitemerr
\newline
\backslash
@latex@warning{Empty `thebibliography' environment}}
\backslash
endlist}
\backslash
def
\latex default
cite#1#1
\latex latex
\backslash
def
\latex default
lbibitem[#1]#2
\layout Standard
\latex latex
\backslash
if
\latex default
@filesw
\latex latex
\backslash
def
\backslash
protect
\latex default
##1
\latex latex
\backslash
string
\latex default
##1
\latex latex
\backslash
space{}
\backslash
immediate
\newline
\backslash
write
\latex default
auxout
\latex latex
\backslash
string
\backslash
bibcite{#2}{#1}
\backslash
fi
\backslash
ignorespaces{}
\backslash
makeatother
\newline
\layout 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.
\latex latex
\backslash
dots
\newline
\layout Section
Fixed-Period Problems: The Sublinear Case
\layout Standard
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_inset Formula
\begin{eqnarray*}
\dot{x} & = & JH'(t,x)\\
x(0) & = & x(T)
\end{eqnarray*}
\end_inset
with
\begin_inset Formula \( H(t,\cdot ) \)
\end_inset
a convex function of
\begin_inset Formula \( x \)
\end_inset
, going to
\begin_inset Formula \( +\infty \)
\end_inset
when
\begin_inset Formula \( \left\Vert x\right\Vert \rightarrow \infty \)
\end_inset
.
\layout Subsection
Autonomous Systems
\layout Standard
In this section, we will consider the case when the Hamiltonian
\begin_inset Formula \( H(x) \)
\end_inset
is autonomous.
For the sake of simplicity, we shall also assume that it is
\begin_inset Formula \( C^{1} \)
\end_inset
.
\layout Standard
We shall first consider the question of nontriviality, within the general
framework of
\begin_inset Formula \( \left( A_{\infty },B_{\infty }\right) \)
\end_inset
-subquadratic Hamiltonians.
In the second subsection, we shall look into the special case when
\begin_inset Formula \( H \)
\end_inset
is
\begin_inset Formula \( \left( 0,b_{\infty }\right) \)
\end_inset
-subquadratic, and we shall try to derive additional information.
\layout Subsubsection
The General Case: Nontriviality.
\layout Standard
We assume that
\begin_inset Formula \( H \)
\end_inset
is
\begin_inset Formula \( \left( A_{\infty },B_{\infty }\right) \)
\end_inset
-sub\SpecialChar \-
qua\SpecialChar \-
dra\SpecialChar \-
tic at infinity, for some constant symmetric matrices
\begin_inset Formula \( A_{\infty } \)
\end_inset
and
\begin_inset Formula \( B_{\infty } \)
\end_inset
, with
\begin_inset Formula \( B_{\infty }-A_{\infty } \)
\end_inset
positive definite.
Set:
\begin_inset Formula
\begin{eqnarray}
\gamma : & = & {\textrm{smallest}\, \, \textrm{eigenvalue}\, \, \textrm{of}}\, \, \,
\, B_{\infty }-A_{\infty }\\
\lambda : & = & {\textrm{largest}\, \, \textrm{negative}\, \, \textrm{eigenvalue}\, \,
\textrm{of}}\, \, J\frac{d}{dt}+A_{\infty }\, \, .
\end{eqnarray}
\end_inset
\layout Standard
Theorem 21 tells us that if
\begin_inset Formula \( \lambda +\gamma <0 \)
\end_inset
, the boundary-value problem:
\begin_inset Formula
\begin{equation}
\begin{array}{rcl}
\dot{x} & = & JH'(x)\\
x(0) & = & x(T)
\end{array}
\end{equation}
\end_inset
has at least one solution
\begin_inset Formula \( \overline{x} \)
\end_inset
, which is found by minimizing the dual action functional:
\begin_inset Formula
\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}
\end_inset
on the range of
\begin_inset Formula \( \Lambda \)
\end_inset
, which is a subspace
\begin_inset Formula \( R(\Lambda )_{L}^{2} \)
\end_inset
with finite codimension.
Here
\begin_inset Formula
\begin{equation}
N(x):=H(x)-\frac{1}{2}\left( A_{\infty }x,x\right)
\end{equation}
\end_inset
is a convex function, and
\begin_inset Formula
\begin{equation}
N(x)\leq \frac{1}{2}\left( \left( B_{\infty }-A_{\infty }\right) x,x\right) +c\, \, \,
\, \, \, \forall x\, \, .
\end{equation}
\end_inset
\layout Proposition
Assume
\begin_inset Formula \( H'(0)=0 \)
\end_inset
and
\begin_inset Formula \( H(0)=0 \)
\end_inset
.
Set:
\begin_inset Formula
\begin{equation}
\label{2eq:one}
\delta :=\liminf _{x\rightarrow 0}2N(x)\left\Vert x\right\Vert ^{-2}\, \, .
\end{equation}
\end_inset
\layout Proposition
If
\begin_inset Formula \( \gamma <-\lambda <\delta \)
\end_inset
, the solution
\begin_inset Formula \( \overline{u} \)
\end_inset
is non-zero:
\begin_inset Formula
\begin{equation}
\overline{x}(t)\ne 0\, \, \, \, \, \, \forall t\, \, .
\end{equation}
\end_inset
\layout Proof
Condition (
\begin_inset LatexCommand \ref{2eq:one}
\end_inset
) means that, for every
\begin_inset Formula \( \delta '>\delta \)
\end_inset
, there is some
\begin_inset Formula \( \varepsilon >0 \)
\end_inset
such that
\begin_inset Formula
\begin{equation}
\left\Vert x\right\Vert \leq \varepsilon \Rightarrow N(x)\leq \frac{\delta
'}{2}\left\Vert x\right\Vert ^{2}\, \, .
\end{equation}
\end_inset
\layout Proof
It is an exercise in convex analysis, into which we shall not go, to show
that this implies that there is an
\begin_inset Formula \( \eta >0 \)
\end_inset
such that
\begin_inset Formula
\begin{equation}
\label{2eq:two}
f\left\Vert x\right\Vert \leq \eta \Rightarrow N^{\ast }(y)\leq \frac{1}{2\delta
'}\left\Vert y\right\Vert ^{2}\, \, .
\end{equation}
\end_inset
\begin_float fig
\layout Proof
\latex latex
\backslash
vspace{2.5cm}
\layout Caption
This is the caption of the figure displaying a white eagle and a white horse
on a snow field
\end_float
\layout Proof
Since
\begin_inset Formula \( u_{1} \)
\end_inset
is a smooth function, we will have
\begin_inset Formula \( \left\Vert hu_{1}\right\Vert _{\infty }\leq \eta \)
\end_inset
for
\begin_inset Formula \( h \)
\end_inset
small enough, and inequality (
\begin_inset LatexCommand \ref{2eq:two}
\end_inset
) will hold, yielding thereby:
\begin_inset Formula
\begin{equation}
\psi (hu_{1})\leq \frac{h^{2}}{2}\frac{1}{\lambda }\left\Vert u_{1}\right\Vert
_{2}^{2}+\frac{h^{2}}{2}\frac{1}{\delta '}\left\Vert u_{1}\right\Vert ^{2}\, \, .
\end{equation}
\end_inset
\layout Proof
If we choose
\begin_inset Formula \( \delta ' \)
\end_inset
close enough to
\begin_inset Formula \( \delta \)
\end_inset
, the quantity
\begin_inset Formula \( \left( \frac{1}{\lambda }+\frac{1}{\delta '}\right) \)
\end_inset
will be negative, and we end up with
\begin_inset Formula
\begin{equation}
\psi (hu_{1})<0\, \, \, \, \, \, \, \, \, \, {\textrm{for}}\, \, \, \, h\ne 0\, \, \,
\, {\textrm{small}}\, \, .
\end{equation}
\end_inset
\layout Proof
On the other hand, we check directly that
\begin_inset Formula \( \psi (0)=0 \)
\end_inset
.
This shows that 0 cannot be a minimizer of
\begin_inset Formula \( \psi \)
\end_inset
, not even a local one.
So
\begin_inset Formula \( \overline{u}\ne 0 \)
\end_inset
and
\begin_inset Formula \( \overline{u}\ne \Lambda _{o}^{-1}(0)=0 \)
\end_inset
.
\latex latex
\backslash
qed
\newline
\layout Corollary
Assume
\begin_inset Formula \( H \)
\end_inset
is
\begin_inset Formula \( C^{2} \)
\end_inset
and
\begin_inset Formula \( \left( a_{\infty },b_{\infty }\right) \)
\end_inset
-subquadratic at infinity.
Let
\begin_inset Formula \( \xi _{1},\allowbreak \dots ,\allowbreak \xi _{N} \)
\end_inset
be the equilibria, that is, the solutions of
\begin_inset Formula \( H'(\xi )=0 \)
\end_inset
.
Denote by
\begin_inset Formula \( \omega _{k} \)
\end_inset
the smallest eigenvalue of
\begin_inset Formula \( H''\left( \xi _{k}\right) \)
\end_inset
, and set:
\begin_inset Formula
\begin{equation}
\omega :={\textrm{Min}\, }\left\{ \omega _{1},\dots ,\omega _{k}\right\} \, \, .
\end{equation}
\end_inset
If:
\begin_inset Formula
\begin{equation}
\label{2eq:three}
\frac{T}{2\pi }b_{\infty }<-E\left[ -\frac{T}{2\pi }a_{\infty }\right] <\frac{T}{2\pi
}\omega
\end{equation}
\end_inset
then minimization of
\begin_inset Formula \( \psi \)
\end_inset
yields a non-constant
\begin_inset Formula \( T \)
\end_inset
-periodic solution
\begin_inset Formula \( \overline{x} \)
\end_inset
.
\layout Standard
We recall once more that by the integer part
\begin_inset Formula \( E[\alpha ] \)
\end_inset
of
\begin_inset Formula \( \alpha \in \bbbr \)
\end_inset
, we mean the
\begin_inset Formula \( a\in \bbbz \)
\end_inset
such that
\begin_inset Formula \( a<\alpha \leq a+1 \)
\end_inset
.
For instance, if we take
\begin_inset Formula \( a_{\infty }=0 \)
\end_inset
, Corollary 2 tells us that
\begin_inset Formula \( \overline{x} \)
\end_inset
exists and is non-constant provided that:
\layout Standard
\begin_inset Formula
\begin{equation}
\frac{T}{2\pi }b_{\infty }<1<\frac{T}{2\pi }
\end{equation}
\end_inset
or
\begin_inset Formula
\begin{equation}
\label{2eq:four}
T\in \left( \frac{2\pi }{\omega },\frac{2\pi }{b_{\infty }}\right) \, \, .
\end{equation}
\end_inset
\layout Proof
The spectrum of
\begin_inset Formula \( \Lambda \)
\end_inset
is
\begin_inset Formula \( \frac{2\pi }{T}\bbbz +a_{\infty } \)
\end_inset
.
The largest negative eigenvalue
\begin_inset Formula \( \lambda \)
\end_inset
is given by
\begin_inset Formula \( \frac{2\pi }{T}k_{o}+a_{\infty } \)
\end_inset
, where
\begin_inset Formula
\begin{equation}
\frac{2\pi }{T}k_{o}+a_{\infty }<0\leq \frac{2\pi }{T}(k_{o}+1)+a_{\infty }\, \, .
\end{equation}
\end_inset
Hence:
\begin_inset Formula
\begin{equation}
k_{o}=E\left[ -\frac{T}{2\pi }a_{\infty }\right] \, \, .
\end{equation}
\end_inset
\layout Proof
The condition
\begin_inset Formula \( \gamma <-\lambda <\delta \)
\end_inset
now becomes:
\begin_inset Formula
\begin{equation}
b_{\infty }-a_{\infty }<-\frac{2\pi }{T}k_{o}-a_{\infty }<\omega -a_{\infty }
\end{equation}
\end_inset
which is precisely condition (
\begin_inset LatexCommand \ref{2eq:three}
\end_inset
).
\latex latex
\backslash
qed
\newline
\layout Lemma
Assume that
\begin_inset Formula \( H \)
\end_inset
is
\begin_inset Formula \( C^{2} \)
\end_inset
on
\begin_inset Formula \( \bbbr ^{2n}\setminus \{0\} \)
\end_inset
and that
\begin_inset Formula \( H''(x) \)
\end_inset
is non-de\SpecialChar \-
gen\SpecialChar \-
er\SpecialChar \-
ate for any
\begin_inset Formula \( x\ne 0 \)
\end_inset
.
Then any local minimizer
\begin_inset Formula \( \widetilde{x} \)
\end_inset
of
\begin_inset Formula \( \psi \)
\end_inset
has minimal period
\begin_inset Formula \( T \)
\end_inset
.
\layout Proof
We know that
\begin_inset Formula \( \widetilde{x} \)
\end_inset
, or
\begin_inset Formula \( \widetilde{x}+\xi \)
\end_inset
for some constant
\begin_inset Formula \( \xi \in \bbbr ^{2n} \)
\end_inset
, is a
\begin_inset Formula \( T \)
\end_inset
-periodic solution of the Hamiltonian system:
\begin_inset Formula
\begin{equation}
\dot{x}=JH'(x)\, \, .
\end{equation}
\end_inset
\layout Proof
There is no loss of generality in taking
\begin_inset Formula \( \xi =0 \)
\end_inset
.
So
\begin_inset Formula \( \psi (x)\geq \psi (\widetilde{x}) \)
\end_inset
for all
\begin_inset Formula \( \widetilde{x} \)
\end_inset
in some neighbourhood of
\begin_inset Formula \( x \)
\end_inset
in
\begin_inset Formula \( W^{1,2}\left( \bbbr /T\bbbz ;\bbbr ^{2n}\right) \)
\end_inset
.
\layout Proof
But this index is precisely the index
\begin_inset Formula \( i_{T}(\widetilde{x}) \)
\end_inset
of the
\begin_inset Formula \( T \)
\end_inset
-periodic solution
\begin_inset Formula \( \widetilde{x} \)
\end_inset
over the interval
\begin_inset Formula \( (0,T) \)
\end_inset
, as defined in Sect.
\protected_separator
2.6.
So
\begin_inset Formula
\begin{equation}
\label{2eq:five}
i_{T}(\widetilde{x})=0\, \, .
\end{equation}
\end_inset
\layout Proof
Now if
\begin_inset Formula \( \widetilde{x} \)
\end_inset
has a lower period,
\begin_inset Formula \( T/k \)
\end_inset
say, we would have, by Corollary 31:
\begin_inset Formula
\begin{equation}
i_{T}(\widetilde{x})=i_{kT/k}(\widetilde{x})\geq ki_{T/k}(\widetilde{x})+k-1\geq
k-1\geq 1\, \, .
\end{equation}
\end_inset
\layout Proof
This would contradict (
\begin_inset LatexCommand \ref{2eq:five}
\end_inset
), and thus cannot happen.
\latex latex
\backslash
qed
\newline
\layout Paragraph
Notes and Comments.
\layout Standard
The results in this section are a refined version of
\begin_inset LatexCommand \cite{2clar:eke}
\end_inset
; the minimality result of Proposition 14 was the first of its kind.
\layout Standard
To understand the nontriviality conditions, such as the one in formula (
\begin_inset LatexCommand \ref{2eq:four}
\end_inset
), one may think of a one-parameter family
\begin_inset Formula \( x_{T} \)
\end_inset
,
\begin_inset Formula \( T\in \left( 2\pi \omega ^{-1},2\pi b_{\infty }^{-1}\right) \)
\end_inset
of periodic solutions,
\begin_inset Formula \( x_{T}(0)=x_{T}(T) \)
\end_inset
, with
\begin_inset Formula \( x_{T} \)
\end_inset
going away to infinity when
\begin_inset Formula \( T\rightarrow 2\pi \omega ^{-1} \)
\end_inset
, which is the period of the linearized system at 0.
\begin_float tab
\layout Caption
This is the example table taken out of
\shape italic
The TeXbook,
\shape default
p.
\latex latex
\backslash
,
\latex default
246
\layout Standard
\align center \LyXTable
multicol5
6 3 0 0 0 0 0 0
1 1 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 1 0 0
8 0 0 "" "r@{\quad}"
4 0 0 "" ""
2 0 0 "" ""
1 2 1 1 0 0 0 "" ""
1 2 1 1 0 0 0 "" ""
2 2 1 1 0 0 0 "" ""
0 8 0 0 0 0 0 "" ""
0 4 0 0 0 0 0 "" ""
0 2 0 0 0 0 0 "" ""
0 8 0 0 0 0 0 "" ""
0 4 0 0 0 0 0 "" ""
0 2 0 0 0 0 0 "" ""
0 8 0 0 0 0 0 "" ""
0 4 0 0 0 0 0 "" ""
0 2 0 0 0 0 0 "" ""
0 8 0 0 0 0 0 "" ""
0 4 0 0 0 0 0 "" ""
0 2 0 0 0 0 0 "" ""
0 8 0 1 0 0 0 "" ""
0 4 0 1 0 0 0 "" ""
0 2 0 1 0 0 0 "" ""
\latex latex
\backslash
rule{0pt}{12pt}
\latex default
Year
\newline
World population
\newline
\latex latex
\backslash
rule{0pt}{12pt}
\latex default
8000 B.C.
\newline
5,000,000
\newline
\newline
50 A.D.
\newline
200,000,000
\newline
\newline
1650 A.D.
\newline
500,000,000
\newline
\newline
1945 A.D.
\newline
2,300,000,000
\newline
\newline
1980 A.D.
\newline
4,400,000,000
\newline
\end_float
\layout Theorem
[Ghoussoub-Preiss] Assume
\begin_inset Formula \( H(t,x) \)
\end_inset
is
\begin_inset Formula \( (0,\varepsilon ) \)
\end_inset
-subquadratic at infinity for all
\begin_inset Formula \( \varepsilon >0 \)
\end_inset
, and
\begin_inset Formula \( T \)
\end_inset
-periodic in
\begin_inset Formula \( t \)
\end_inset
\begin_inset Formula
\begin{equation}
H(t,\cdot )\, \, \, \, \, \, \, \, \, \, {\textrm{is}\, \, \textrm{convex}}\, \, \, \,
\forall t
\end{equation}
\end_inset
\begin_inset Formula
\begin{equation}
H(\cdot ,x)\, \, \, \, \, \, \, \, \, \, {\textrm{is}}\, \, \, \,
T{-\textrm{periodic}}\, \, \, \, \forall x
\end{equation}
\end_inset
\begin_inset Formula
\begin{equation}
H(t,x)\geq n\left( \left\Vert x\right\Vert \right) \, \, \, \, \, \, \, \,
{\textrm{with}}\, \, \, \, n(s)s^{-1}\rightarrow \infty \, \, \, \, {\textrm{as}}\, \,
\, \, s\rightarrow \infty
\end{equation}
\end_inset
\begin_inset Formula
\begin{equation}
\forall \varepsilon >0\, \, ,\, \, \, \, \, \, \exists c\, \, :H(t,x)\leq
\frac{\varepsilon }{2}\left\Vert x\right\Vert ^{2}+c\, \, .
\end{equation}
\end_inset
\layout Theorem
Assume also that
\begin_inset Formula \( H \)
\end_inset
is
\begin_inset Formula \( C^{2} \)
\end_inset
, and
\begin_inset Formula \( H''(t,x) \)
\end_inset
is positive definite everywhere.
Then there is a sequence
\begin_inset Formula \( x_{k} \)
\end_inset
,
\begin_inset Formula \( k\in \bbbn \)
\end_inset
, of
\begin_inset Formula \( kT \)
\end_inset
-periodic solutions of the system
\begin_inset Formula
\begin{equation}
\dot{x}=JH'(t,x)
\end{equation}
\end_inset
such that, for every
\begin_inset Formula \( k\in \bbbn \)
\end_inset
, there is some
\begin_inset Formula \( p_{o}\in \bbbn \)
\end_inset
with:
\begin_inset Formula
\begin{equation}
p\geq p_{o}\Rightarrow x_{pk}\ne x_{k}\, \, .
\end{equation}
\end_inset
\latex latex
\backslash
qed
\newline
\layout Example
[
\family roman
External forcing
\family default
] Consider the system:
\begin_inset Formula
\begin{equation}
\dot{x}=JH'(x)+f(t)
\end{equation}
\end_inset
where the Hamiltonian
\begin_inset Formula \( H \)
\end_inset
is
\begin_inset Formula \( \left( 0,b_{\infty }\right) \)
\end_inset
-subquadratic, and the forcing term is a distribution on the circle:
\begin_inset Formula
\begin{equation}
f=\frac{d}{dt}F+f_{o}\, \, \, \, \, \, \, \, {\textrm{with}}\, \, \, \, F\in
L^{2}\left( \bbbr /T\bbbz ;\bbbr ^{2n}\right) \, \, ,
\end{equation}
\end_inset
where
\begin_inset Formula \( f_{o}:=T^{-1}\int _{o}^{T}f(t)dt \)
\end_inset
.
For instance,
\begin_inset Formula
\begin{equation}
f(t)=\sum _{k\in \bbbn }\delta _{k}\xi \, \, ,
\end{equation}
\end_inset
where
\begin_inset Formula \( \delta _{k} \)
\end_inset
is the Dirac mass at
\begin_inset Formula \( t=k \)
\end_inset
and
\begin_inset Formula \( \xi \in \bbbr ^{2n} \)
\end_inset
is a constant, fits the prescription.
This means that the system
\begin_inset Formula \( \dot{x}=JH'(x) \)
\end_inset
is being excited by a series of identical shocks at interval
\begin_inset Formula \( T \)
\end_inset
.
\layout Definition
Let
\begin_inset Formula \( A_{\infty }(t) \)
\end_inset
and
\begin_inset Formula \( B_{\infty }(t) \)
\end_inset
be symmetric operators in
\begin_inset Formula \( \bbbr ^{2n} \)
\end_inset
, depending continuously on
\begin_inset Formula \( t\in [0,T] \)
\end_inset
, such that
\begin_inset Formula \( A_{\infty }(t)\leq B_{\infty }(t) \)
\end_inset
for all
\begin_inset Formula \( t \)
\end_inset
.
\layout Definition
A Borelian function
\begin_inset Formula \( H:[0,T]\times \bbbr ^{2n}\rightarrow \bbbr \)
\end_inset
is called
\begin_inset Formula \( \left( A_{\infty },B_{\infty }\right) \)
\end_inset
-
\shape italic
subquadratic at infinity
\shape default
if there exists a function
\begin_inset Formula \( N(t,x) \)
\end_inset
such that:
\begin_inset Formula
\begin{equation}
H(t,x)=\frac{1}{2}\left( A_{\infty }(t)x,x\right) +N(t,x)
\end{equation}
\end_inset
\begin_inset Formula
\begin{equation}
\forall t\, \, ,\, \, \, \, \, \, N(t,x)\, \, \, \, \, \, \, \, {\textrm{is}\, \,
\textrm{convex}\, \, \textrm{with}\, \, \textrm{ respect}\, \, \textrm{ to}}\, \, \,
\, x
\end{equation}
\end_inset
\begin_inset Formula
\begin{equation}
N(t,x)\geq n\left( \left\Vert x\right\Vert \right) \, \, \, \, \, \, \, \,
{\textrm{with}}\, \, \, \, n(s)s^{-1}\rightarrow +\infty \, \, \, \, {\textrm{as}}\,
\, \, \, s\rightarrow +\infty
\end{equation}
\end_inset
\begin_inset Formula
\begin{equation}
\exists c\in \bbbr \, \, :\, \, \, \, \, \, H(t,x)\leq \frac{1}{2}\left( B_{\infty
}(t)x,x\right) +c\, \, \, \, \, \, \forall x\, \, .
\end{equation}
\end_inset
\layout Definition
If
\begin_inset Formula \( A_{\infty }(t)=a_{\infty }I \)
\end_inset
and
\begin_inset Formula \( B_{\infty }(t)=b_{\infty }I \)
\end_inset
, with
\begin_inset Formula \( a_{\infty }\leq b_{\infty }\in \bbbr \)
\end_inset
, we shall say that
\begin_inset Formula \( H \)
\end_inset
is
\begin_inset Formula \( \left( a_{\infty },b_{\infty }\right) \)
\end_inset
-subquadratic at infinity.
As an example, the function
\begin_inset Formula \( \left\Vert x\right\Vert ^{\alpha } \)
\end_inset
, with
\begin_inset Formula \( 1\leq \alpha <2 \)
\end_inset
, is
\begin_inset Formula \( (0,\varepsilon ) \)
\end_inset
-subquadratic at infinity for every
\begin_inset Formula \( \varepsilon >0 \)
\end_inset
.
Similarly, the Hamiltonian
\begin_inset Formula
\begin{equation}
H(t,x)=\frac{1}{2}k\left\Vert k\right\Vert ^{2}+\left\Vert x\right\Vert ^{\alpha }
\end{equation}
\end_inset
is
\begin_inset Formula \( (k,k+\varepsilon ) \)
\end_inset
-subquadratic for every
\begin_inset Formula \( \varepsilon >0 \)
\end_inset
.
Note that, if
\begin_inset Formula \( k<0 \)
\end_inset
, it is not convex.
\layout Paragraph
Notes and Comments.
\layout Standard
The first results on subharmonics were obtained by Rabinowitz in
\begin_inset LatexCommand \cite{2rab}
\end_inset
, who showed the existence of infinitely many subharmonics both in the subquadra
tic and superquadratic case, with suitable growth conditions on
\begin_inset Formula \( H' \)
\end_inset
.
Again the duality approach enabled Clarke and Ekeland in
\begin_inset LatexCommand \cite{2clar:eke:2}
\end_inset
to treat the same problem in the convex-subquadratic case, with growth
conditions on
\begin_inset Formula \( H \)
\end_inset
only.
\layout Standard
Recently, Michalek and Tarantello (see Michalek, R., Tarantello, G.
\begin_inset LatexCommand \cite{2mich:tar}
\end_inset
and Tarantello, G.
\begin_inset LatexCommand \cite{2tar}
\end_inset
) have obtained lower bound on the number of subharmonics of period
\begin_inset Formula \( kT \)
\end_inset
, based on symmetry considerations and on pinching estimates, as in Sect.
\protected_separator
5.2 of this article.
\layout Standard
\latex latex
\backslash
begin{thebibliography}{}
\newline
\backslash
bibitem[1980]{2clar:eke}
\newline
Clarke, F., Ekeland, I.:
\newline
Nonlinear oscillations and
\newline
boundary-value problems for Hamiltonian systems.
\newline
Arch.
Rat.
Mech.
Anal.
{
\backslash
textbf{78}} (1982) 315--333
\newline
\newline
\backslash
bibitem[1981]{2clar:eke:2}
\newline
Clarke, F., Ekeland, I.:
\newline
Solutions p
\backslash
'{e}riodiques, du
\newline
p
\backslash
'{e}riode donn
\backslash
'{e}e, des
\backslash
'{e}quations hamiltoniennes.
\newline
Note CRAS Paris {
\backslash
textbf{287}} (1978) 1013--1015
\newline
\newline
\backslash
bibitem[1982]{2mich:tar}
\newline
Michalek, R., Tarantello, G.:
\newline
Subharmonic solutions with prescribed minimal
\newline
period for nonautonomous Hamiltonian systems.
\newline
J.
Diff.
Eq.
{
\backslash
textbf{72}} (1988) 28--55
\newline
\newline
\backslash
bibitem[1983]{2tar}
\newline
Tarantello, G.:
\newline
Subharmonic solutions for Hamiltonian
\newline
systems via a
\backslash
(
\backslash
bbbz_{p}
\backslash
) pseudoindex theory.
\newline
Annali di Matematica Pura (to appear)
\newline
\newline
\backslash
bibitem[1985]{2rab}
\newline
Rabinowitz, P.:
\newline
On subharmonic solutions of a Hamiltonian system.
\newline
Comm.
Pure Appl.
Math.
{
\backslash
textbf{33}} (1980) 609--633
\newline
\newline
\backslash
end{thebibliography}
\backslash
clearpage
\newline
\backslash
addtocmark[2]{Author Index}
\backslash
renewcommand{
\backslash
indexname}{Author Index}
\latex default
\begin_inset LatexCommand \printindex{}
\end_inset
\latex latex
\backslash
clearpage
\newline
\backslash
addtocmark[2]{Subject Index}
\backslash
markboth{Subject Index}{
\latex default
Subject Index
\latex latex
}
\backslash
renewcommand{
\backslash
indexname}{Subject Index}
\latex default
\begin_inset Include \input{subjidx.ind}
\end_inset
\the_end
