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\def\12{{1\ov 2}}
\def\al{\alpha}
\def\Aun{A_\un}
\def\aun{a_\un}
\def\bullet{\cdot}
\def\Bun{B_\un}
\def\bun{b_\un}
\def\de{\delta}
\def\dx{\dot x}
\def\ep{\epsilon}
\def\fa{\forall}
\def\for{{\rm for}}
\def\Lai{\Lambda}
\def\lb{\left[}
\def\lg{\left\{}
\def\liminfuu{{\rm lim inf}$\,$}
\def\liminfu{\mathop{\vphantom{\tst\sum}\hbox{\liminfuu}}}
\def\limsupuu{{\rm lim sup}$\,$}
\def\limsupu{\mathop{\vphantom{\tst\sum}\hbox{\limsupuu}}}
\def\lr{\left(}
\def\lss{\left\|}
\def\Min{{\rm Min\,}}
\def\NN{\bbbn}
\def\ol{\overline}
\def\om{\omega}
\def\ov{\over}
\def\rb{\right]}
\def\rg{\right\}}
\def\RRn{\bbbr^{2n}}
\def\RR{\bbbr}
\def\rr{\right)}
\def\rss{\right\|}
\def\sm{\setminus}
\def\tst{\textstyle}
\def\tx{\wt x}
\def\un{\infty}
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\contribution{Hamiltonian Mechanics}
\author{Ivar Ekeland@1 and Roger Temam@2}
\address{@1Princeton University, Princeton, NJ 08544, USA
@2Universit\'e de Paris-Sud, Laboratoire d'Analyse Num\'erique,
B\^atiment 425, F-91405 Orsay Cedex, France}
\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}
\titlea{1}{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
$$\eqalign{\dot x &= JH' (t,x)\cr x(0) &= x(T)\cr}$$
with $H(t,\bullet )$ a convex function of $x$, going to $+\un$ when
$\lss x\rss \to \un$.

\titleb{1.1}{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 $\lr \Aun , \Bun\rr$-subquadratic Hamiltonians. In
the second subsection, we shall look into the special case when $H$ is
$\lr 0,\bun\rr$-subquadratic, and we shall try to derive additional
information.
\titlec{ The General Case: Nontriviality.}
We assume that $H$ is $\lr \Aun , \Bun \rr$-sub\-qua\-dra\-tic at
infinity, for some constant symmetric matrices $\Aun$ and $\Bun$, with
$\Bun -\Aun$ positive definite. Set:
$$\eqalignno{
\gamma :& = {\rm smallest\ eigenvalue\  of}\ \ \Bun - \Aun  & (1)\cr
\lambda : & = {\rm largest\ negative\ eigenvalue\ of}\ \ J {d\ov dt}
+\Aun\ . & (2)\cr}$$

Theorem 21 tells us that if $\lambda +\gamma < 0$, the boundary-value
problem:
$$\eqalign{ \dx &= JH' (x)\cr
x(0) &= x (T)\cr}\eqno(3)$$
has at least one solution $\ol x$, which is found by minimizing the dual
action functional:
$$ \psi (u) = \int_o^T \lb \12 \lr \Lai_o^{-1} u,u\rr + N^\ast (-u)\rb
dt\eqno(4)$$
on the range of $\Lai$, which is a subspace $R (\Lai )\sb L^2$ with
finite codimension. Here
$$ N(x) := H(x) - \12 \lr \Aun x,x\rr\eqno(5)$$
is a convex function, and
$$ N(x) \le \12 \lr \lr \Bun - \Aun\rr x,x\rr + c\ \ \ \fa x\
.\eqno(6)$$

\proposition{ 1.} { Assume $H'(0)=0$ and $ H(0)=0$. Set:
$$ \de := \liminfu_{x\to 0} 2 N (x) \lss x\rss^{-2}\ .\eqno(7)$$

If $\gamma < - \lambda < \de$, the solution $\ol u$ is non-zero:
$$ \ol x (t) \ne 0\ \ \ \fa t\ .\eqno(8)$$}
\proof{} Condition (7) means that, for every $\de ' > \de$, there is
some $\ep > 0$ such that
$$ \lss x\rss \le \ep \Rightarrow N (x) \le {\de '\ov 2} \lss x\rss^2\
.\eqno(9)$$

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
$$ f\lss x\rss \le \eta \Rightarrow N^\ast (y) \le {1\ov 2\de '} \lss
y\rss^2\ .\eqno(10)$$

\begfig 2.5cm
\figure{1}{This is the caption of the figure displaying a white eagle
and a white horse on a snow field}
\endfig

Since $u_1$ is a smooth function, we will have $\lss hu_1\rss_\un \le
\eta$ for $h$ small enough, and inequality (10) will hold, yielding
thereby:
$$ \psi (hu_1) \le {h^2\ov 2} {1\ov \lambda} \lss u_1 \rss_2^2 + {h^2\ov
2} {1\ov \de '} \lss u_1\rss^2\ .\eqno(11)$$

If we choose $\de '$ close enough to $\de$, the quantity $\lr {1\ov
\lambda} + {1\ov \de '}\rr$ will be negative, and we end up with $$ \psi
(hu_1) < 0\ \ \ \ \ \for\ \ h\ne 0\ \ {\rm small}\ .\eqno(12)$$

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 $\ol u
\ne 0$ and $\ol u \ne \Lai_o^{-1} (0) = 0$. \qed

\corollary{ 2.} { Assume $H$ is $C^2$ and $\lr \aun
,\bun\rr$-subquadratic at infinity. Let
$\xi_1,\allowbreak\dots,\allowbreak\xi_N$  be the
equilibria, that is, the solutions of $H' (\xi ) = 0$. Denote by $\om_k$
the smallest eigenvalue of $H'' \lr \xi_k\rr$, and set:
$$ \om : = \Min \lg \om_1 , \dots , \om_k\rg\ .\eqno(13)$$
If:
$$ {T\ov 2\pi} \bun < - E \lb - {T\ov 2\pi}\aun\rb < {T\ov
2\pi}\om\eqno(14)$$
then minimization of $\psi$ yields a non-constant $T$-periodic solution
$\ol x$.}
We recall once more that by the integer part $E [\al ]$ of $\al \in
\RR$, we mean the $a\in \ZZ$ such that $a< \al \le a+1$. For instance,
if we take $\aun = 0$, Corollary 2 tells us that $\ol x$ exists and is
non-constant provided that:
$$ {T\ov 2\pi} \bun < 1 < {T\ov 2\pi}\eqno(15)$$
or
$$ T\in \lr {2\pi\ov \om},{2\pi\ov \bun}\rr\ .\eqno(16)$$
\proof{} The spectrum of $\Lai$ is ${2\pi\ov T} \ZZ +\aun$. The largest
negative eigenvalue $\lambda$ is given by ${2\pi\ov T}k_o +\aun$, where
$$ {2\pi\ov T}k_o + \aun < 0\le {2\pi\ov T} (k_o +1) + \aun\
.\eqno(17)$$
Hence:
$$ k_o = E \lb - {T\ov 2\pi} \aun\rb \ .\eqno(18)$$

The condition $\gamma < -\lambda < \de$ now becomes:
$$ \bun - \aun < - {2\pi\ov T} k_o -\aun < \om -\aun\eqno(19)$$
which is precisely condition (14).\qed

\lemma {3.} { Assume that $H$ is $C^2$ on $\RRn \sm \{ 0\}$ and
that $H'' (x)$ is non-degenerate for any $x\ne 0$. Then any local
minimizer $\tx$ of $\psi$ has minimal period $T$.}
\proof{} We know that $\tx$, or $\tx + \xi$ for some constant $\xi
\in \RRn$, is a $T$-periodic solution of the Hamiltonian system:
$$ \dx = JH' (x)\ .\eqno(20)$$

There is no loss of generality in taking $\xi = 0$. So $\psi (x) \ge
\psi (\tx )$ for all $\tx$ in some neighbourhood of $x$ in $W^{1,2} \lr
\RR / T\ZZ ; \RRn\rr$.

But this index is precisely the index $i_T (\tx )$ of the $T$-periodic
solution $\tx$ over the interval $(0,T)$, as defined in Sect.~2.6. So
$$ i_T (\tx ) = 0\ .\eqno(21)$$

Now if $\tx$ has a lower period, $T/k$ say, we would have, by Corollary
31:
$$ i_T (\tx ) = i_{kT/k}(\tx ) \ge ki_{T/k} (\tx ) + k-1 \ge k-1 \ge
1\ .\eqno(22)$$

This would contradict (21), and thus cannot happen.\qed
\titled{Notes and Comments.} The results in this section are a
refined
version of [1]; the minimality result of Proposition 14 was the first
of its kind.

To understand the nontriviality conditions, such as the one in formula
(16), one may think of a one-parameter family $x_T$, $T\in \lr
2\pi\om^{-1}, 2\pi \bun^{-1}\rr$ of periodic solutions, $x_T (0) = x_T
(T)$, with $x_T$ going away to infinity when $T\to 2\pi \om^{-1}$, which
is the period of the linearized system at 0.

\vskip8 true mm
\tabcap{1}{This is the example table took out of the
\TeX{} book page 246}
\vbox{\petit\hrule\smallskip
\halign{\hfil#\quad&\quad#\hfil\cr
Year\hfill&World population\cr
\noalign{\smallskip\hrule\smallskip}
8000 B.C.&\phantom{1,00}5,000,000\cr
50 A.D.&\phantom{1,}200,000,000\cr
1650 A.D.&\phantom{1,}500,000,000\cr
1945 A.D.&2,300,000,000\cr
1980 A.D.&4,400,000,000\cr}
\smallskip\hrule}
\vskip 8 true mm

\theorem{4 (Ghoussoub-Preiss).} { Assume $H(t,x)$ is
$(0,\ep )$-subquadratic at
infinity for all $\ep > 0$, and $T$-periodic in $t$
$$ H (t,\bullet )\ \ \ \ \ {\rm is\ convex}\ \ \fa t\eqno(23)$$
$$ H (\bullet ,x)\ \ \ \ \ {\rm is}\ \ T{\rm -periodic}\ \ \fa x
\eqno(24)$$
$$ H (t,x)\ge n\lr \lss x\rss\rr\ \ \ \ \ {\rm with}\ \ n (s)s^{-1}\to
\un\ \ {\rm as}\ \ s\to \un\eqno(25)$$
$$ \fa \ep > 0\ ,\ \ \ \exists c\ :\ H(t,x) \le {\ep\ov 2}\lss x\rss^2 +
c\ .\eqno(26)$$

Assume also that $H$ is $C^2$, and $H'' (t,x)$ is positive definite
everywhere. Then there is a sequence $x_k$, $k\in \NN$, of $kT$-periodic
solutions of the system
$$ \dx = JH' (t,x)\eqno(27)$$
such that, for every $k\in \NN$, there is some $p_o\in\NN$ with:
$$ p\ge p_o\Rightarrow x_{pk} \ne x_k\ .\eqno(28)$$\qed}
\example {1 {\rm(External forcing).}}{ Consider the system:
$$ \dx = JH' (x) + f(t)\eqno(29)$$
where the Hamiltonian $H$ is $\lr 0,\bun\rr$-subquadratic, and the
forcing term is a distribution on the circle:
$$ f = {d\ov dt} F + f_o\ \ \ \ \ {\rm with}\ \ F\in L^2 \lr \RR / T\ZZ
; \RRn\rr\ ,\eqno(30)$$
where $f_o : = T^{-1}\int_o^T f (t) dt$. For instance,
$$ f (t) = \sum_{k\in \NN} \de_k \xi\ ,\eqno(31)$$
where $\de_k$ is the Dirac mass at $t= k$ and $\xi \in \RRn$ is a
constant, fits the prescription. This means that the system $\dx = JH'
(x)$ is being excited by a series of identical shocks at interval $T$.}

\definition{5.}{Let $A_\un (t)$ and $B_\un (t)$ be symmetric
operators in $\RRn$, depending continuously on $t\in [0,T]$, such that
$A_\un (t) \le B_\un (t)$ for all $t$.

A Borelian function $H: [0,T]\times \RRn \to \RR$ is called $\lr A_\un
,B_\un\rr$-{\it subquadratic at infinity} if there exists a function
$N(t,x)$ such that:
$$ H (t,x) = \12 \lr A_\un (t) x,x\rr + N(t,x)\eqno(32)$$
$$ \fa t\ ,\ \ \ N(t,x)\ \ \ \ \ {\rm is\ convex\ with\  respect\  to}\
\ x\eqno(33)$$
$$ N(t,x) \ge n\lr \lss x\rss\rr\ \ \ \ \ {\rm with}\ \ n(s)s^{-1}\to
+\un\ \ {\rm as}\ \ s\to +\un\eqno(34)$$
$$ \exists c\in \RR\ :\ \ \ H (t,x) \le \12 \lr B_\un (t) x,x\rr + c\ \
\ \fa x\ .\eqno(35)$$
}
If $A_\un (t) = a_\un I$ and $B_\un (t) = b_\un I$, with $a_\un \le
b_\un \in \RR$, we shall say that $H$ is $\lr a_\un
,b_\un\rr$-subquadratic at infinity. As an example, the function $\lss x
\rss^\al$, with $1\le \al < 2$, is $(0,\ep )$-subquadratic at infinity
for every $\ep > 0$. Similarly, the Hamiltonian
$$ H (t,x) = \12 k \lss k\rss^2 +\lss x\rss^\al\eqno(36)$$
is $(k,k+\ep )$-subquadratic for every $\ep > 0$. Note that, if $k<0$,
it is not convex.

\titled{Notes and Comments.} The first results on subharmonics were
obtained by Rabinowitz in [5], 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 [2] to treat the same problem in the
convex-subquadratic case, with growth conditions on $H$ only.

Recently, Michalek and Tarantello (see [3] and [4]) 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.
\begref{References}{5.}
\refno {1.} Clarke, F., Ekeland, I.: Nonlinear oscillations and
boundary-value problems for Hamiltonian systems. Arch. Rat. Mech. Anal.
{\bf 78} (1982) 315--333
\refno {2.} Clarke, F., Ekeland, I.: Solutions p\'eriodiques, du
p\'eriode donn\'ee, des \'equations hamiltoiennes. Note CRAS Paris {\bf
287} (1978) 1013--1015
\refno {3.} Michalek, R., Tarantello, G.: Subharmonic solutions with
prescribed minimal period for nonautonomous Hamiltonian systems. J.
Diff. Eq. {\bf 72} (1988) 28--55
\refno {4.} Tarantello, G.: Subharmonic solutions for Hamiltonian
systems via a $\bbbz_p$ pseudoindex theory. Annali di Mathematica Pura
(to appear)
\refno {5.} Rabinowitz, P.: On subharmonic solutions of a
Hamiltonian system. Comm. Pure Appl. Math. {\bf 33} (1980) 609--633
\endref
\byebye