paper: remove all but a single article

miscellaneous/paper/sozopol13/proceedings.tex

@@ -9,162 +9,6 @@
 %
 \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.
-
-\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
@@ -610,484 +454,4 @@ 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
-%
-% Modify the bibliography environment to call for the author-year
-% system. This is done normally with the citeauthoryear option
-% for a particular contribution.
-\makeatletter
-\renewenvironment{thebibliography}[1]
-     {\section*{\refname}
-      \small
-      \list{}%
-           {\settowidth\labelwidth{}%
-            \leftmargin\parindent
-            \itemindent=-\parindent
-            \labelsep=\z@
-            \if@openbib
-              \advance\leftmargin\bibindent
-              \itemindent -\bibindent
-              \listparindent \itemindent
-              \parsep \z@
-            \fi
-            \usecounter{enumiv}%
-            \let\p@enumiv\@empty
-            \renewcommand\theenumiv{}}%
-      \if@openbib
-        \renewcommand\newblock{\par}%
-      \else
-        \renewcommand\newblock{\hskip .11em \@plus.33em \@minus.07em}%
-      \fi
-      \sloppy\clubpenalty4000\widowpenalty4000%
-      \sfcode`\.=\@m}
-     {\def\@noitemerr
-       {\@latex@warning{Empty `thebibliography' environment}}%
-      \endlist}
-      \def\@cite#1{#1}%
-      \def\@lbibitem[#1]#2{\item[]\if@filesw
-        {\def\protect##1{\string ##1\space}\immediate
-      \write\@auxout{\string\bibcite{#2}{#1}}}\fi\ignorespaces}
-\makeatother
-%
-\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)
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