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- % This is PLNCS.DEM the demonstration file of
- % the plain TeX macro package from Springer-Verlag
- % for Lecture Notes in Computer Science, version 1.1
- \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}
- \def\wt{\widetilde}
- \def\ZZ{\bbbz}
- \input plncs.cmm
- \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