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\begin{document}
	\mainmatter              % start of a contribution
	%
	\title{Hamiltonian Mechanics}
	%
	\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 200 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
		% We would like to encourage you to list your keywords within
		% the abstract section using the \keywords{...} command.
		\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}[!ht]
			% Use the relevant command for your figure-insertion program
			% to insert the figure file.
			% For example, with the graphicx style use
			\begin{center}
				\includegraphics[scale=.65]{figure}
			\end{center}
			%
			% If no graphics program available, insert a blank space i.e. use
			%\picplace{5cm}{2cm} % Give the correct figure height and width in cm
			%
			\caption{Example of  \texttt{includegraphics} command to include figures}
			\label{fig:1}       % Give a unique label
		\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{smit:wat};
	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{fo:kes:nic:tue}, 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{may:ehr:stein} to treat the
same problem in the convex-subquadratic case, with growth conditions on
$H$ only.

Recently, Michalek and Tarantello (see \cite{czaj:fitz} and \cite{fost:kes} )
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.


\subsubsection*{Acknowledgements}
If you want to include acknowledgments of assistance and the like at the end of an individual chapter.


%
% ---- Bibliography ----
%
\begin{thebibliography}{6}
	%
	
		\bibitem {czaj:fitz}
	Czajkowski, K., Fitzgerald, S., Foster, I., Kesselman, C.: Grid information services
	for distributed resource sharing. In: 10th IEEE International Symposium
	on High Performance Distributed Computing, pp. 181--184. IEEE Press, New York (2001).
	\url{doi: 10.1109/HPDC.2001.945188}
	
	\bibitem {fost:kes}
	Foster, I., Kesselman, C.: The Grid: Blueprint for a New Computing Infrastructure.
	Morgan Kaufmann, San Francisco (1999)
	
	
	\bibitem {fo:kes:nic:tue}
	Foster, I., Kesselman, C., Nick, J., Tuecke, S.: The physiology of the grid: an open grid services architecture for distributed systems integration. Technical report, Global Grid Forum (2002)
	
	\bibitem {may:ehr:stein}
	May, P., Ehrlich, H.-C., Steinke, T.: ZIB structure prediction pipeline:
	composing a complex biological workflow through web services.
	In: Nagel, W.E., Walter, W.V., Lehner, W. (eds.) Euro-Par 2006.
	LNCS, vol. 4128, pp. 1148--1158. Springer, Heidelberg (2006).
	\url{doi:10.1007/11823285_121}
	
	
	\bibitem {onlyurl}
	National Center for Biotechnology Information. \url{http://www.ncbi.nlm.nih.gov}
	
	\bibitem {smit:wat}
	Smith, T.F., Waterman, M.S.: Identification of common molecular subsequences.
	J. Mol. Biol. 147, 195--197 (1981). \url{doi:10.1016/0022-2836(81)90087-5}
	
	
\end{thebibliography}

\end{document}
	
%
\section*{Appendix}
\addcontentsline{toc}{section}{Appendix}
%
%
Troughout the body of the manuscript references may be cited  by author-year  using  \verb"natbib.sty" package  as in the following examples:

\begin{enumerate}
	\item \citet{Broy2002} or \citet{Dod1999} for a Contribution;
	\item \citet{Hamburger1995} in a Journal article \citep[see also][]{SlifkaWhitton2000};
	\item  \citet{Geddesetal1992} in a Monograph.
	
\end{enumerate}


Where there are three or more authors, acknowledge only
the first author, e.g., Geddes et al. (1992). For two authors the
following style should be used: Slifka and Whitton (2000).



In mathematical texts references are often labelled as author-year acronyms. In order to achieve this simply give an optional argument to the \verb|\bibitem| command. Always use \verb|\bibitem| and \verb|\cite| for cross-referencing.

\medskip

When producing your bibliography please make sure that the data is complete (name and initial of author, year of publication, book title, publisher's name and place, journal name, volume number, page numbers) and up to date (e.g. edition number).


If there are several works by the same author, the following order should be used:
\begin{enumerate}
	\item all works by the author alone, ordered chronologically by year of publication
	\item all works by the author with a coauthor, ordered alphabetically by coauthor
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\end{enumerate}

Always use the standard abbreviation of a journal's name according to the ISSN \textit{List of Title Word Abbreviations}.



Please make sure that all references from the list are cited in the text. 


\medskip

% BibTeX users please use
% \bibliography{}
%
\bibliographystyle{spbasic}

\begin{thebibliography}{6}
	
	% Contribution 
	\bibitem[Broy(2002)]{Broy2002} Broy, M. (2002). Software engineering --- from auxiliary to key technologies. In: Broy, M., Dener, E. (eds.) Software Pioneers, pp. 10-13. Springer, Heidelberg 
	%
	% Online Document
	\bibitem[Dod(1999)]{Dod1999} Dod, J. (1999). Effective substances. In: The Dictionary of Substances and Their Effects. Royal Society of Chemistry.  Available via DIALOG. \\
	\url{http://www.rsc.org/dose/title of subordinate documen}. Cited 15 Jan 1999
	%
	% Monograph
	\bibitem[Geddes et al.(1992)]{Geddesetal1992} Geddes, K.O., Czapor, S.R., Labahn, G. (1992).  Algorithms for Computer Algebra. Kluwer, Boston 
	%
	% Journal article
	\bibitem[Hamburger(1995)]{Hamburger1995} Hamburger, C. (1995). Quasimonotonicity, regularity and duality for nonlinear systems of partial differential equations. Ann. Mat. Pura. Appl. \textbf{169}, 321--354 
	%
	% Journal article by DOI
	\bibitem[Slifka and Whitton(2000)]{SlifkaWhitton2000} Slifka, M.K., Whitton, J.L. (2000) Clinical implications of dysregulated cytokine production. J. Mol. Med. \url{ 10.1007/s001090000086} 
	
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