llncs.dem (35361B)
1 % This is LLNCS.DEM the demonstration file of 2 % the LaTeX macro package from Springer-Verlag 3 % for Lecture Notes in Computer Science, 4 % version 2.4 for LaTeX2e as of 16. April 2010 5 % 6 \documentclass{llncs} 7 % 8 \usepackage{makeidx} % allows for indexgeneration 9 % 10 \begin{document} 11 % 12 \frontmatter % for the preliminaries 13 % 14 \pagestyle{headings} % switches on printing of running heads 15 \addtocmark{Hamiltonian Mechanics} % additional mark in the TOC 16 % 17 \chapter*{Preface} 18 % 19 This textbook is intended for use by students of physics, physical 20 chemistry, and theoretical chemistry. The reader is presumed to have a 21 basic knowledge of atomic and quantum physics at the level provided, for 22 example, by the first few chapters in our book {\it The Physics of Atoms 23 and Quanta}. The student of physics will find here material which should 24 be included in the basic education of every physicist. This book should 25 furthermore allow students to acquire an appreciation of the breadth and 26 variety within the field of molecular physics and its future as a 27 fascinating area of research. 28 29 For the student of chemistry, the concepts introduced in this book will 30 provide a theoretical framework for that entire field of study. With the 31 help of these concepts, it is at least in principle possible to reduce 32 the enormous body of empirical chemical knowledge to a few basic 33 principles: those of quantum mechanics. In addition, modern physical 34 methods whose fundamentals are introduced here are becoming increasingly 35 important in chemistry and now represent indispensable tools for the 36 chemist. As examples, we might mention the structural analysis of 37 complex organic compounds, spectroscopic investigation of very rapid 38 reaction processes or, as a practical application, the remote detection 39 of pollutants in the air. 40 41 \vspace{1cm} 42 \begin{flushright}\noindent 43 April 1995\hfill Walter Olthoff\\ 44 Program Chair\\ 45 ECOOP'95 46 \end{flushright} 47 % 48 \chapter*{Organization} 49 ECOOP'95 is organized by the department of Computer Science, Univeristy 50 of \AA rhus and AITO (association Internationa pour les Technologie 51 Object) in cooperation with ACM/SIGPLAN. 52 % 53 \section*{Executive Commitee} 54 \begin{tabular}{@{}p{5cm}@{}p{7.2cm}@{}} 55 Conference Chair:&Ole Lehrmann Madsen (\AA rhus University, DK)\\ 56 Program Chair: &Walter Olthoff (DFKI GmbH, Germany)\\ 57 Organizing Chair:&J\o rgen Lindskov Knudsen (\AA rhus University, DK)\\ 58 Tutorials:&Birger M\o ller-Pedersen\hfil\break 59 (Norwegian Computing Center, Norway)\\ 60 Workshops:&Eric Jul (University of Kopenhagen, Denmark)\\ 61 Panels:&Boris Magnusson (Lund University, Sweden)\\ 62 Exhibition:&Elmer Sandvad (\AA rhus University, DK)\\ 63 Demonstrations:&Kurt N\o rdmark (\AA rhus University, DK) 64 \end{tabular} 65 % 66 \section*{Program Commitee} 67 \begin{tabular}{@{}p{5cm}@{}p{7.2cm}@{}} 68 Conference Chair:&Ole Lehrmann Madsen (\AA rhus University, DK)\\ 69 Program Chair: &Walter Olthoff (DFKI GmbH, Germany)\\ 70 Organizing Chair:&J\o rgen Lindskov Knudsen (\AA rhus University, DK)\\ 71 Tutorials:&Birger M\o ller-Pedersen\hfil\break 72 (Norwegian Computing Center, Norway)\\ 73 Workshops:&Eric Jul (University of Kopenhagen, Denmark)\\ 74 Panels:&Boris Magnusson (Lund University, Sweden)\\ 75 Exhibition:&Elmer Sandvad (\AA rhus University, DK)\\ 76 Demonstrations:&Kurt N\o rdmark (\AA rhus University, DK) 77 \end{tabular} 78 % 79 \begin{multicols}{3}[\section*{Referees}] 80 V.~Andreev\\ 81 B\"arwolff\\ 82 E.~Barrelet\\ 83 H.P.~Beck\\ 84 G.~Bernardi\\ 85 E.~Binder\\ 86 P.C.~Bosetti\\ 87 Braunschweig\\ 88 F.W.~B\"usser\\ 89 T.~Carli\\ 90 A.B.~Clegg\\ 91 G.~Cozzika\\ 92 S.~Dagoret\\ 93 Del~Buono\\ 94 P.~Dingus\\ 95 H.~Duhm\\ 96 J.~Ebert\\ 97 S.~Eichenberger\\ 98 R.J.~Ellison\\ 99 Feltesse\\ 100 W.~Flauger\\ 101 A.~Fomenko\\ 102 G.~Franke\\ 103 J.~Garvey\\ 104 M.~Gennis\\ 105 L.~Goerlich\\ 106 P.~Goritchev\\ 107 H.~Greif\\ 108 E.M.~Hanlon\\ 109 R.~Haydar\\ 110 R.C.W.~Henderso\\ 111 P.~Hill\\ 112 H.~Hufnagel\\ 113 A.~Jacholkowska\\ 114 Johannsen\\ 115 S.~Kasarian\\ 116 I.R.~Kenyon\\ 117 C.~Kleinwort\\ 118 T.~K\"ohler\\ 119 S.D.~Kolya\\ 120 P.~Kostka\\ 121 U.~Kr\"uger\\ 122 J.~Kurzh\"ofer\\ 123 M.P.J.~Landon\\ 124 A.~Lebedev\\ 125 Ch.~Ley\\ 126 F.~Linsel\\ 127 H.~Lohmand\\ 128 Martin\\ 129 S.~Masson\\ 130 K.~Meier\\ 131 C.A.~Meyer\\ 132 S.~Mikocki\\ 133 J.V.~Morris\\ 134 B.~Naroska\\ 135 Nguyen\\ 136 U.~Obrock\\ 137 G.D.~Patel\\ 138 Ch.~Pichler\\ 139 S.~Prell\\ 140 F.~Raupach\\ 141 V.~Riech\\ 142 P.~Robmann\\ 143 N.~Sahlmann\\ 144 P.~Schleper\\ 145 Sch\"oning\\ 146 B.~Schwab\\ 147 A.~Semenov\\ 148 G.~Siegmon\\ 149 J.R.~Smith\\ 150 M.~Steenbock\\ 151 U.~Straumann\\ 152 C.~Thiebaux\\ 153 P.~Van~Esch\\ 154 from Yerevan Ph\\ 155 L.R.~West\\ 156 G.-G.~Winter\\ 157 T.P.~Yiou\\ 158 M.~Zimmer\end{multicols} 159 % 160 \section*{Sponsoring Institutions} 161 % 162 Bernauer-Budiman Inc., Reading, Mass.\\ 163 The Hofmann-International Company, San Louis Obispo, Cal.\\ 164 Kramer Industries, Heidelberg, Germany 165 % 166 \tableofcontents 167 % 168 \mainmatter % start of the contributions 169 % 170 \title{Hamiltonian Mechanics unter besonderer Ber\"ucksichtigung der 171 h\"ohreren Lehranstalten} 172 % 173 \titlerunning{Hamiltonian Mechanics} % abbreviated title (for running head) 174 % also used for the TOC unless 175 % \toctitle is used 176 % 177 \author{Ivar Ekeland\inst{1} \and Roger Temam\inst{2} 178 Jeffrey Dean \and David Grove \and Craig Chambers \and Kim~B.~Bruce \and 179 Elsa Bertino} 180 % 181 \authorrunning{Ivar Ekeland et al.} % abbreviated author list (for running head) 182 % 183 %%%% list of authors for the TOC (use if author list has to be modified) 184 \tocauthor{Ivar Ekeland, Roger Temam, Jeffrey Dean, David Grove, 185 Craig Chambers, Kim B. Bruce, and Elisa Bertino} 186 % 187 \institute{Princeton University, Princeton NJ 08544, USA,\\ 188 \email{I.Ekeland@princeton.edu},\\ WWW home page: 189 \texttt{http://users/\homedir iekeland/web/welcome.html} 190 \and 191 Universit\'{e} de Paris-Sud, 192 Laboratoire d'Analyse Num\'{e}rique, B\^{a}timent 425,\\ 193 F-91405 Orsay Cedex, France} 194 195 \maketitle % typeset the title of the contribution 196 197 \begin{abstract} 198 The abstract should summarize the contents of the paper 199 using at least 70 and at most 150 words. It will be set in 9-point 200 font size and be inset 1.0 cm from the right and left margins. 201 There will be two blank lines before and after the Abstract. \dots 202 \keywords{computational geometry, graph theory, Hamilton cycles} 203 \end{abstract} 204 % 205 \section{Fixed-Period Problems: The Sublinear Case} 206 % 207 With this chapter, the preliminaries are over, and we begin the search 208 for periodic solutions to Hamiltonian systems. All this will be done in 209 the convex case; that is, we shall study the boundary-value problem 210 \begin{eqnarray*} 211 \dot{x}&=&JH' (t,x)\\ 212 x(0) &=& x(T) 213 \end{eqnarray*} 214 with $H(t,\cdot)$ a convex function of $x$, going to $+\infty$ when 215 $\left\|x\right\| \to \infty$. 216 217 % 218 \subsection{Autonomous Systems} 219 % 220 In this section, we will consider the case when the Hamiltonian $H(x)$ 221 is autonomous. For the sake of simplicity, we shall also assume that it 222 is $C^{1}$. 223 224 We shall first consider the question of nontriviality, within the 225 general framework of 226 $\left(A_{\infty},B_{\infty}\right)$-subquadratic Hamiltonians. In 227 the second subsection, we shall look into the special case when $H$ is 228 $\left(0,b_{\infty}\right)$-subquadratic, 229 and we shall try to derive additional information. 230 % 231 \subsubsection{The General Case: Nontriviality.} 232 % 233 We assume that $H$ is 234 $\left(A_{\infty},B_{\infty}\right)$-sub\-qua\-dra\-tic at infinity, 235 for some constant symmetric matrices $A_{\infty}$ and $B_{\infty}$, 236 with $B_{\infty}-A_{\infty}$ positive definite. Set: 237 \begin{eqnarray} 238 \gamma :&=&{\rm smallest\ eigenvalue\ of}\ \ B_{\infty} - A_{\infty} \\ 239 \lambda : &=& {\rm largest\ negative\ eigenvalue\ of}\ \ 240 J \frac{d}{dt} +A_{\infty}\ . 241 \end{eqnarray} 242 243 Theorem~\ref{ghou:pre} tells us that if $\lambda +\gamma < 0$, the 244 boundary-value problem: 245 \begin{equation} 246 \begin{array}{rcl} 247 \dot{x}&=&JH' (x)\\ 248 x(0)&=&x (T) 249 \end{array} 250 \end{equation} 251 has at least one solution 252 $\overline{x}$, which is found by minimizing the dual 253 action functional: 254 \begin{equation} 255 \psi (u) = \int_{o}^{T} \left[\frac{1}{2} 256 \left(\Lambda_{o}^{-1} u,u\right) + N^{\ast} (-u)\right] dt 257 \end{equation} 258 on the range of $\Lambda$, which is a subspace $R (\Lambda)_{L}^{2}$ 259 with finite codimension. Here 260 \begin{equation} 261 N(x) := H(x) - \frac{1}{2} \left(A_{\infty} x,x\right) 262 \end{equation} 263 is a convex function, and 264 \begin{equation} 265 N(x) \le \frac{1}{2} 266 \left(\left(B_{\infty} - A_{\infty}\right) x,x\right) 267 + c\ \ \ \forall x\ . 268 \end{equation} 269 270 % 271 \begin{proposition} 272 Assume $H'(0)=0$ and $ H(0)=0$. Set: 273 \begin{equation} 274 \delta := \liminf_{x\to 0} 2 N (x) \left\|x\right\|^{-2}\ . 275 \label{eq:one} 276 \end{equation} 277 278 If $\gamma < - \lambda < \delta$, 279 the solution $\overline{u}$ is non-zero: 280 \begin{equation} 281 \overline{x} (t) \ne 0\ \ \ \forall t\ . 282 \end{equation} 283 \end{proposition} 284 % 285 \begin{proof} 286 Condition (\ref{eq:one}) means that, for every 287 $\delta ' > \delta$, there is some $\varepsilon > 0$ such that 288 \begin{equation} 289 \left\|x\right\| \le \varepsilon \Rightarrow N (x) \le 290 \frac{\delta '}{2} \left\|x\right\|^{2}\ . 291 \end{equation} 292 293 It is an exercise in convex analysis, into which we shall not go, to 294 show that this implies that there is an $\eta > 0$ such that 295 \begin{equation} 296 f\left\|x\right\| \le \eta 297 \Rightarrow N^{\ast} (y) \le \frac{1}{2\delta '} 298 \left\|y\right\|^{2}\ . 299 \label{eq:two} 300 \end{equation} 301 302 \begin{figure} 303 \vspace{2.5cm} 304 \caption{This is the caption of the figure displaying a white eagle and 305 a white horse on a snow field} 306 \end{figure} 307 308 Since $u_{1}$ is a smooth function, we will have 309 $\left\|hu_{1}\right\|_\infty \le \eta$ 310 for $h$ small enough, and inequality (\ref{eq:two}) will hold, 311 yielding thereby: 312 \begin{equation} 313 \psi (hu_{1}) \le \frac{h^{2}}{2} 314 \frac{1}{\lambda} \left\|u_{1} \right\|_{2}^{2} + \frac{h^{2}}{2} 315 \frac{1}{\delta '} \left\|u_{1}\right\|^{2}\ . 316 \end{equation} 317 318 If we choose $\delta '$ close enough to $\delta$, the quantity 319 $\left(\frac{1}{\lambda} + \frac{1}{\delta '}\right)$ 320 will be negative, and we end up with 321 \begin{equation} 322 \psi (hu_{1}) < 0\ \ \ \ \ {\rm for}\ \ h\ne 0\ \ {\rm small}\ . 323 \end{equation} 324 325 On the other hand, we check directly that $\psi (0) = 0$. This shows 326 that 0 cannot be a minimizer of $\psi$, not even a local one. 327 So $\overline{u} \ne 0$ and 328 $\overline{u} \ne \Lambda_{o}^{-1} (0) = 0$. \qed 329 \end{proof} 330 % 331 \begin{corollary} 332 Assume $H$ is $C^{2}$ and 333 $\left(a_{\infty},b_{\infty}\right)$-subquadratic at infinity. Let 334 $\xi_{1},\allowbreak\dots,\allowbreak\xi_{N}$ be the 335 equilibria, that is, the solutions of $H' (\xi ) = 0$. 336 Denote by $\omega_{k}$ 337 the smallest eigenvalue of $H'' \left(\xi_{k}\right)$, and set: 338 \begin{equation} 339 \omega : = {\rm Min\,} \left\{\omega_{1},\dots,\omega_{k}\right\}\ . 340 \end{equation} 341 If: 342 \begin{equation} 343 \frac{T}{2\pi} b_{\infty} < 344 - E \left[- \frac{T}{2\pi}a_{\infty}\right] < 345 \frac{T}{2\pi}\omega 346 \label{eq:three} 347 \end{equation} 348 then minimization of $\psi$ yields a non-constant $T$-periodic solution 349 $\overline{x}$. 350 \end{corollary} 351 % 352 353 We recall once more that by the integer part $E [\alpha ]$ of 354 $\alpha \in \bbbr$, we mean the $a\in \bbbz$ 355 such that $a< \alpha \le a+1$. For instance, 356 if we take $a_{\infty} = 0$, Corollary 2 tells 357 us that $\overline{x}$ exists and is 358 non-constant provided that: 359 360 \begin{equation} 361 \frac{T}{2\pi} b_{\infty} < 1 < \frac{T}{2\pi} 362 \end{equation} 363 or 364 \begin{equation} 365 T\in \left(\frac{2\pi}{\omega},\frac{2\pi}{b_{\infty}}\right)\ . 366 \label{eq:four} 367 \end{equation} 368 369 % 370 \begin{proof} 371 The spectrum of $\Lambda$ is $\frac{2\pi}{T} \bbbz +a_{\infty}$. The 372 largest negative eigenvalue $\lambda$ is given by 373 $\frac{2\pi}{T}k_{o} +a_{\infty}$, 374 where 375 \begin{equation} 376 \frac{2\pi}{T}k_{o} + a_{\infty} < 0 377 \le \frac{2\pi}{T} (k_{o} +1) + a_{\infty}\ . 378 \end{equation} 379 Hence: 380 \begin{equation} 381 k_{o} = E \left[- \frac{T}{2\pi} a_{\infty}\right] \ . 382 \end{equation} 383 384 The condition $\gamma < -\lambda < \delta$ now becomes: 385 \begin{equation} 386 b_{\infty} - a_{\infty} < 387 - \frac{2\pi}{T} k_{o} -a_{\infty} < \omega -a_{\infty} 388 \end{equation} 389 which is precisely condition (\ref{eq:three}).\qed 390 \end{proof} 391 % 392 393 \begin{lemma} 394 Assume that $H$ is $C^{2}$ on $\bbbr^{2n} \setminus \{ 0\}$ and 395 that $H'' (x)$ is non-de\-gen\-er\-ate for any $x\ne 0$. Then any local 396 minimizer $\widetilde{x}$ of $\psi$ has minimal period $T$. 397 \end{lemma} 398 % 399 \begin{proof} 400 We know that $\widetilde{x}$, or 401 $\widetilde{x} + \xi$ for some constant $\xi 402 \in \bbbr^{2n}$, is a $T$-periodic solution of the Hamiltonian system: 403 \begin{equation} 404 \dot{x} = JH' (x)\ . 405 \end{equation} 406 407 There is no loss of generality in taking $\xi = 0$. So 408 $\psi (x) \ge \psi (\widetilde{x} )$ 409 for all $\widetilde{x}$ in some neighbourhood of $x$ in 410 $W^{1,2} \left(\bbbr / T\bbbz ; \bbbr^{2n}\right)$. 411 412 But this index is precisely the index 413 $i_{T} (\widetilde{x} )$ of the $T$-periodic 414 solution $\widetilde{x}$ over the interval 415 $(0,T)$, as defined in Sect.~2.6. So 416 \begin{equation} 417 i_{T} (\widetilde{x} ) = 0\ . 418 \label{eq:five} 419 \end{equation} 420 421 Now if $\widetilde{x}$ has a lower period, $T/k$ say, 422 we would have, by Corollary 31: 423 \begin{equation} 424 i_{T} (\widetilde{x} ) = 425 i_{kT/k}(\widetilde{x} ) \ge 426 ki_{T/k} (\widetilde{x} ) + k-1 \ge k-1 \ge 1\ . 427 \end{equation} 428 429 This would contradict (\ref{eq:five}), and thus cannot happen.\qed 430 \end{proof} 431 % 432 \paragraph{Notes and Comments.} 433 The results in this section are a 434 refined version of \cite{clar:eke}; 435 the minimality result of Proposition 436 14 was the first of its kind. 437 438 To understand the nontriviality conditions, such as the one in formula 439 (\ref{eq:four}), one may think of a one-parameter family 440 $x_{T}$, $T\in \left(2\pi\omega^{-1}, 2\pi b_{\infty}^{-1}\right)$ 441 of periodic solutions, $x_{T} (0) = x_{T} (T)$, 442 with $x_{T}$ going away to infinity when $T\to 2\pi \omega^{-1}$, 443 which is the period of the linearized system at 0. 444 445 \begin{table} 446 \caption{This is the example table taken out of {\it The 447 \TeX{}book,} p.\,246} 448 \begin{center} 449 \begin{tabular}{r@{\quad}rl} 450 \hline 451 \multicolumn{1}{l}{\rule{0pt}{12pt} 452 Year}&\multicolumn{2}{l}{World population}\\[2pt] 453 \hline\rule{0pt}{12pt} 454 8000 B.C. & 5,000,000& \\ 455 50 A.D. & 200,000,000& \\ 456 1650 A.D. & 500,000,000& \\ 457 1945 A.D. & 2,300,000,000& \\ 458 1980 A.D. & 4,400,000,000& \\[2pt] 459 \hline 460 \end{tabular} 461 \end{center} 462 \end{table} 463 % 464 \begin{theorem} [Ghoussoub-Preiss]\label{ghou:pre} 465 Assume $H(t,x)$ is 466 $(0,\varepsilon )$-subquadratic at 467 infinity for all $\varepsilon > 0$, and $T$-periodic in $t$ 468 \begin{equation} 469 H (t,\cdot )\ \ \ \ \ {\rm is\ convex}\ \ \forall t 470 \end{equation} 471 \begin{equation} 472 H (\cdot ,x)\ \ \ \ \ {\rm is}\ \ T{\rm -periodic}\ \ \forall x 473 \end{equation} 474 \begin{equation} 475 H (t,x)\ge n\left(\left\|x\right\|\right)\ \ \ \ \ 476 {\rm with}\ \ n (s)s^{-1}\to \infty\ \ {\rm as}\ \ s\to \infty 477 \end{equation} 478 \begin{equation} 479 \forall \varepsilon > 0\ ,\ \ \ \exists c\ :\ 480 H(t,x) \le \frac{\varepsilon}{2}\left\|x\right\|^{2} + c\ . 481 \end{equation} 482 483 Assume also that $H$ is $C^{2}$, and $H'' (t,x)$ is positive definite 484 everywhere. Then there is a sequence $x_{k}$, $k\in \bbbn$, of 485 $kT$-periodic solutions of the system 486 \begin{equation} 487 \dot{x} = JH' (t,x) 488 \end{equation} 489 such that, for every $k\in \bbbn$, there is some $p_{o}\in\bbbn$ with: 490 \begin{equation} 491 p\ge p_{o}\Rightarrow x_{pk} \ne x_{k}\ . 492 \end{equation} 493 \qed 494 \end{theorem} 495 % 496 \begin{example} [{{\rm External forcing}}] 497 Consider the system: 498 \begin{equation} 499 \dot{x} = JH' (x) + f(t) 500 \end{equation} 501 where the Hamiltonian $H$ is 502 $\left(0,b_{\infty}\right)$-subquadratic, and the 503 forcing term is a distribution on the circle: 504 \begin{equation} 505 f = \frac{d}{dt} F + f_{o}\ \ \ \ \ 506 {\rm with}\ \ F\in L^{2} \left(\bbbr / T\bbbz; \bbbr^{2n}\right)\ , 507 \end{equation} 508 where $f_{o} : = T^{-1}\int_{o}^{T} f (t) dt$. For instance, 509 \begin{equation} 510 f (t) = \sum_{k\in \bbbn} \delta_{k} \xi\ , 511 \end{equation} 512 where $\delta_{k}$ is the Dirac mass at $t= k$ and 513 $\xi \in \bbbr^{2n}$ is a 514 constant, fits the prescription. This means that the system 515 $\dot{x} = JH' (x)$ is being excited by a 516 series of identical shocks at interval $T$. 517 \end{example} 518 % 519 \begin{definition} 520 Let $A_{\infty} (t)$ and $B_{\infty} (t)$ be symmetric 521 operators in $\bbbr^{2n}$, depending continuously on 522 $t\in [0,T]$, such that 523 $A_{\infty} (t) \le B_{\infty} (t)$ for all $t$. 524 525 A Borelian function 526 $H: [0,T]\times \bbbr^{2n} \to \bbbr$ 527 is called 528 $\left(A_{\infty} ,B_{\infty}\right)$-{\it subquadratic at infinity} 529 if there exists a function $N(t,x)$ such that: 530 \begin{equation} 531 H (t,x) = \frac{1}{2} \left(A_{\infty} (t) x,x\right) + N(t,x) 532 \end{equation} 533 \begin{equation} 534 \forall t\ ,\ \ \ N(t,x)\ \ \ \ \ 535 {\rm is\ convex\ with\ respect\ to}\ \ x 536 \end{equation} 537 \begin{equation} 538 N(t,x) \ge n\left(\left\|x\right\|\right)\ \ \ \ \ 539 {\rm with}\ \ n(s)s^{-1}\to +\infty\ \ {\rm as}\ \ s\to +\infty 540 \end{equation} 541 \begin{equation} 542 \exists c\in \bbbr\ :\ \ \ H (t,x) \le 543 \frac{1}{2} \left(B_{\infty} (t) x,x\right) + c\ \ \ \forall x\ . 544 \end{equation} 545 546 If $A_{\infty} (t) = a_{\infty} I$ and 547 $B_{\infty} (t) = b_{\infty} I$, with 548 $a_{\infty} \le b_{\infty} \in \bbbr$, 549 we shall say that $H$ is 550 $\left(a_{\infty},b_{\infty}\right)$-subquadratic 551 at infinity. As an example, the function 552 $\left\|x\right\|^{\alpha}$, with 553 $1\le \alpha < 2$, is $(0,\varepsilon )$-subquadratic at infinity 554 for every $\varepsilon > 0$. Similarly, the Hamiltonian 555 \begin{equation} 556 H (t,x) = \frac{1}{2} k \left\|k\right\|^{2} +\left\|x\right\|^{\alpha} 557 \end{equation} 558 is $(k,k+\varepsilon )$-subquadratic for every $\varepsilon > 0$. 559 Note that, if $k<0$, it is not convex. 560 \end{definition} 561 % 562 563 \paragraph{Notes and Comments.} 564 The first results on subharmonics were 565 obtained by Rabinowitz in \cite{rab}, who showed the existence of 566 infinitely many subharmonics both in the subquadratic and superquadratic 567 case, with suitable growth conditions on $H'$. Again the duality 568 approach enabled Clarke and Ekeland in \cite{clar:eke:2} to treat the 569 same problem in the convex-subquadratic case, with growth conditions on 570 $H$ only. 571 572 Recently, Michalek and Tarantello (see \cite{mich:tar} and \cite{tar}) 573 have obtained lower bound on the number of subharmonics of period $kT$, 574 based on symmetry considerations and on pinching estimates, as in 575 Sect.~5.2 of this article. 576 577 % 578 % ---- Bibliography ---- 579 % 580 \begin{thebibliography}{5} 581 % 582 \bibitem {clar:eke} 583 Clarke, F., Ekeland, I.: 584 Nonlinear oscillations and 585 boundary-value problems for Hamiltonian systems. 586 Arch. Rat. Mech. Anal. 78, 315--333 (1982) 587 588 \bibitem {clar:eke:2} 589 Clarke, F., Ekeland, I.: 590 Solutions p\'{e}riodiques, du 591 p\'{e}riode donn\'{e}e, des \'{e}quations hamiltoniennes. 592 Note CRAS Paris 287, 1013--1015 (1978) 593 594 \bibitem {mich:tar} 595 Michalek, R., Tarantello, G.: 596 Subharmonic solutions with prescribed minimal 597 period for nonautonomous Hamiltonian systems. 598 J. Diff. Eq. 72, 28--55 (1988) 599 600 \bibitem {tar} 601 Tarantello, G.: 602 Subharmonic solutions for Hamiltonian 603 systems via a $\bbbz_{p}$ pseudoindex theory. 604 Annali di Matematica Pura (to appear) 605 606 \bibitem {rab} 607 Rabinowitz, P.: 608 On subharmonic solutions of a Hamiltonian system. 609 Comm. Pure Appl. Math. 33, 609--633 (1980) 610 611 \end{thebibliography} 612 613 % 614 % second contribution with nearly identical text, 615 % slightly changed contribution head (all entries 616 % appear as defaults), and modified bibliography 617 % 618 \title{Hamiltonian Mechanics2} 619 620 \author{Ivar Ekeland\inst{1} \and Roger Temam\inst{2}} 621 622 \institute{Princeton University, Princeton NJ 08544, USA 623 \and 624 Universit\'{e} de Paris-Sud, 625 Laboratoire d'Analyse Num\'{e}rique, B\^{a}timent 425,\\ 626 F-91405 Orsay Cedex, France} 627 628 \maketitle 629 % 630 % Modify the bibliography environment to call for the author-year 631 % system. This is done normally with the citeauthoryear option 632 % for a particular contribution. 633 \makeatletter 634 \renewenvironment{thebibliography}[1] 635 {\section*{\refname} 636 \small 637 \list{}% 638 {\settowidth\labelwidth{}% 639 \leftmargin\parindent 640 \itemindent=-\parindent 641 \labelsep=\z@ 642 \if@openbib 643 \advance\leftmargin\bibindent 644 \itemindent -\bibindent 645 \listparindent \itemindent 646 \parsep \z@ 647 \fi 648 \usecounter{enumiv}% 649 \let\p@enumiv\@empty 650 \renewcommand\theenumiv{}}% 651 \if@openbib 652 \renewcommand\newblock{\par}% 653 \else 654 \renewcommand\newblock{\hskip .11em \@plus.33em \@minus.07em}% 655 \fi 656 \sloppy\clubpenalty4000\widowpenalty4000% 657 \sfcode`\.=\@m} 658 {\def\@noitemerr 659 {\@latex@warning{Empty `thebibliography' environment}}% 660 \endlist} 661 \def\@cite#1{#1}% 662 \def\@lbibitem[#1]#2{\item[]\if@filesw 663 {\def\protect##1{\string ##1\space}\immediate 664 \write\@auxout{\string\bibcite{#2}{#1}}}\fi\ignorespaces} 665 \makeatother 666 % 667 \begin{abstract} 668 The abstract should summarize the contents of the paper 669 using at least 70 and at most 150 words. It will be set in 9-point 670 font size and be inset 1.0 cm from the right and left margins. 671 There will be two blank lines before and after the Abstract. \dots 672 \keywords{graph transformations, convex geometry, lattice computations, 673 convex polygons, triangulations, discrete geometry} 674 \end{abstract} 675 % 676 \section{Fixed-Period Problems: The Sublinear Case} 677 % 678 With this chapter, the preliminaries are over, and we begin the search 679 for periodic solutions to Hamiltonian systems. All this will be done in 680 the convex case; that is, we shall study the boundary-value problem 681 \begin{eqnarray*} 682 \dot{x}&=&JH' (t,x)\\ 683 x(0) &=& x(T) 684 \end{eqnarray*} 685 with $H(t,\cdot)$ a convex function of $x$, going to $+\infty$ when 686 $\left\|x\right\| \to \infty$. 687 688 % 689 \subsection{Autonomous Systems} 690 % 691 In this section, we will consider the case when the Hamiltonian $H(x)$ 692 is autonomous. For the sake of simplicity, we shall also assume that it 693 is $C^{1}$. 694 695 We shall first consider the question of nontriviality, within the 696 general framework of 697 $\left(A_{\infty},B_{\infty}\right)$-subquadratic Hamiltonians. In 698 the second subsection, we shall look into the special case when $H$ is 699 $\left(0,b_{\infty}\right)$-subquadratic, 700 and we shall try to derive additional information. 701 % 702 \subsubsection{The General Case: Nontriviality.} 703 % 704 We assume that $H$ is 705 $\left(A_{\infty},B_{\infty}\right)$-sub\-qua\-dra\-tic at infinity, 706 for some constant symmetric matrices $A_{\infty}$ and $B_{\infty}$, 707 with $B_{\infty}-A_{\infty}$ positive definite. Set: 708 \begin{eqnarray} 709 \gamma :&=&{\rm smallest\ eigenvalue\ of}\ \ B_{\infty} - A_{\infty} \\ 710 \lambda : &=& {\rm largest\ negative\ eigenvalue\ of}\ \ 711 J \frac{d}{dt} +A_{\infty}\ . 712 \end{eqnarray} 713 714 Theorem 21 tells us that if $\lambda +\gamma < 0$, the boundary-value 715 problem: 716 \begin{equation} 717 \begin{array}{rcl} 718 \dot{x}&=&JH' (x)\\ 719 x(0)&=&x (T) 720 \end{array} 721 \end{equation} 722 has at least one solution 723 $\overline{x}$, which is found by minimizing the dual 724 action functional: 725 \begin{equation} 726 \psi (u) = \int_{o}^{T} \left[\frac{1}{2} 727 \left(\Lambda_{o}^{-1} u,u\right) + N^{\ast} (-u)\right] dt 728 \end{equation} 729 on the range of $\Lambda$, which is a subspace $R (\Lambda)_{L}^{2}$ 730 with finite codimension. Here 731 \begin{equation} 732 N(x) := H(x) - \frac{1}{2} \left(A_{\infty} x,x\right) 733 \end{equation} 734 is a convex function, and 735 \begin{equation} 736 N(x) \le \frac{1}{2} 737 \left(\left(B_{\infty} - A_{\infty}\right) x,x\right) 738 + c\ \ \ \forall x\ . 739 \end{equation} 740 741 % 742 \begin{proposition} 743 Assume $H'(0)=0$ and $ H(0)=0$. Set: 744 \begin{equation} 745 \delta := \liminf_{x\to 0} 2 N (x) \left\|x\right\|^{-2}\ . 746 \label{2eq:one} 747 \end{equation} 748 749 If $\gamma < - \lambda < \delta$, 750 the solution $\overline{u}$ is non-zero: 751 \begin{equation} 752 \overline{x} (t) \ne 0\ \ \ \forall t\ . 753 \end{equation} 754 \end{proposition} 755 % 756 \begin{proof} 757 Condition (\ref{2eq:one}) means that, for every 758 $\delta ' > \delta$, there is some $\varepsilon > 0$ such that 759 \begin{equation} 760 \left\|x\right\| \le \varepsilon \Rightarrow N (x) \le 761 \frac{\delta '}{2} \left\|x\right\|^{2}\ . 762 \end{equation} 763 764 It is an exercise in convex analysis, into which we shall not go, to 765 show that this implies that there is an $\eta > 0$ such that 766 \begin{equation} 767 f\left\|x\right\| \le \eta 768 \Rightarrow N^{\ast} (y) \le \frac{1}{2\delta '} 769 \left\|y\right\|^{2}\ . 770 \label{2eq:two} 771 \end{equation} 772 773 \begin{figure} 774 \vspace{2.5cm} 775 \caption{This is the caption of the figure displaying a white eagle and 776 a white horse on a snow field} 777 \end{figure} 778 779 Since $u_{1}$ is a smooth function, we will have 780 $\left\|hu_{1}\right\|_\infty \le \eta$ 781 for $h$ small enough, and inequality (\ref{2eq:two}) will hold, 782 yielding thereby: 783 \begin{equation} 784 \psi (hu_{1}) \le \frac{h^{2}}{2} 785 \frac{1}{\lambda} \left\|u_{1} \right\|_{2}^{2} + \frac{h^{2}}{2} 786 \frac{1}{\delta '} \left\|u_{1}\right\|^{2}\ . 787 \end{equation} 788 789 If we choose $\delta '$ close enough to $\delta$, the quantity 790 $\left(\frac{1}{\lambda} + \frac{1}{\delta '}\right)$ 791 will be negative, and we end up with 792 \begin{equation} 793 \psi (hu_{1}) < 0\ \ \ \ \ {\rm for}\ \ h\ne 0\ \ {\rm small}\ . 794 \end{equation} 795 796 On the other hand, we check directly that $\psi (0) = 0$. This shows 797 that 0 cannot be a minimizer of $\psi$, not even a local one. 798 So $\overline{u} \ne 0$ and 799 $\overline{u} \ne \Lambda_{o}^{-1} (0) = 0$. \qed 800 \end{proof} 801 % 802 \begin{corollary} 803 Assume $H$ is $C^{2}$ and 804 $\left(a_{\infty},b_{\infty}\right)$-subquadratic at infinity. Let 805 $\xi_{1},\allowbreak\dots,\allowbreak\xi_{N}$ be the 806 equilibria, that is, the solutions of $H' (\xi ) = 0$. 807 Denote by $\omega_{k}$ 808 the smallest eigenvalue of $H'' \left(\xi_{k}\right)$, and set: 809 \begin{equation} 810 \omega : = {\rm Min\,} \left\{\omega_{1},\dots,\omega_{k}\right\}\ . 811 \end{equation} 812 If: 813 \begin{equation} 814 \frac{T}{2\pi} b_{\infty} < 815 - E \left[- \frac{T}{2\pi}a_{\infty}\right] < 816 \frac{T}{2\pi}\omega 817 \label{2eq:three} 818 \end{equation} 819 then minimization of $\psi$ yields a non-constant $T$-periodic solution 820 $\overline{x}$. 821 \end{corollary} 822 % 823 824 We recall once more that by the integer part $E [\alpha ]$ of 825 $\alpha \in \bbbr$, we mean the $a\in \bbbz$ 826 such that $a< \alpha \le a+1$. For instance, 827 if we take $a_{\infty} = 0$, Corollary 2 tells 828 us that $\overline{x}$ exists and is 829 non-constant provided that: 830 831 \begin{equation} 832 \frac{T}{2\pi} b_{\infty} < 1 < \frac{T}{2\pi} 833 \end{equation} 834 or 835 \begin{equation} 836 T\in \left(\frac{2\pi}{\omega},\frac{2\pi}{b_{\infty}}\right)\ . 837 \label{2eq:four} 838 \end{equation} 839 840 % 841 \begin{proof} 842 The spectrum of $\Lambda$ is $\frac{2\pi}{T} \bbbz +a_{\infty}$. The 843 largest negative eigenvalue $\lambda$ is given by 844 $\frac{2\pi}{T}k_{o} +a_{\infty}$, 845 where 846 \begin{equation} 847 \frac{2\pi}{T}k_{o} + a_{\infty} < 0 848 \le \frac{2\pi}{T} (k_{o} +1) + a_{\infty}\ . 849 \end{equation} 850 Hence: 851 \begin{equation} 852 k_{o} = E \left[- \frac{T}{2\pi} a_{\infty}\right] \ . 853 \end{equation} 854 855 The condition $\gamma < -\lambda < \delta$ now becomes: 856 \begin{equation} 857 b_{\infty} - a_{\infty} < 858 - \frac{2\pi}{T} k_{o} -a_{\infty} < \omega -a_{\infty} 859 \end{equation} 860 which is precisely condition (\ref{2eq:three}).\qed 861 \end{proof} 862 % 863 864 \begin{lemma} 865 Assume that $H$ is $C^{2}$ on $\bbbr^{2n} \setminus \{ 0\}$ and 866 that $H'' (x)$ is non-de\-gen\-er\-ate for any $x\ne 0$. Then any local 867 minimizer $\widetilde{x}$ of $\psi$ has minimal period $T$. 868 \end{lemma} 869 % 870 \begin{proof} 871 We know that $\widetilde{x}$, or 872 $\widetilde{x} + \xi$ for some constant $\xi 873 \in \bbbr^{2n}$, is a $T$-periodic solution of the Hamiltonian system: 874 \begin{equation} 875 \dot{x} = JH' (x)\ . 876 \end{equation} 877 878 There is no loss of generality in taking $\xi = 0$. So 879 $\psi (x) \ge \psi (\widetilde{x} )$ 880 for all $\widetilde{x}$ in some neighbourhood of $x$ in 881 $W^{1,2} \left(\bbbr / T\bbbz ; \bbbr^{2n}\right)$. 882 883 But this index is precisely the index 884 $i_{T} (\widetilde{x} )$ of the $T$-periodic 885 solution $\widetilde{x}$ over the interval 886 $(0,T)$, as defined in Sect.~2.6. So 887 \begin{equation} 888 i_{T} (\widetilde{x} ) = 0\ . 889 \label{2eq:five} 890 \end{equation} 891 892 Now if $\widetilde{x}$ has a lower period, $T/k$ say, 893 we would have, by Corollary 31: 894 \begin{equation} 895 i_{T} (\widetilde{x} ) = 896 i_{kT/k}(\widetilde{x} ) \ge 897 ki_{T/k} (\widetilde{x} ) + k-1 \ge k-1 \ge 1\ . 898 \end{equation} 899 900 This would contradict (\ref{2eq:five}), and thus cannot happen.\qed 901 \end{proof} 902 % 903 \paragraph{Notes and Comments.} 904 The results in this section are a 905 refined version of \cite{2clar:eke}; 906 the minimality result of Proposition 907 14 was the first of its kind. 908 909 To understand the nontriviality conditions, such as the one in formula 910 (\ref{2eq:four}), one may think of a one-parameter family 911 $x_{T}$, $T\in \left(2\pi\omega^{-1}, 2\pi b_{\infty}^{-1}\right)$ 912 of periodic solutions, $x_{T} (0) = x_{T} (T)$, 913 with $x_{T}$ going away to infinity when $T\to 2\pi \omega^{-1}$, 914 which is the period of the linearized system at 0. 915 916 \begin{table} 917 \caption{This is the example table taken out of {\it The 918 \TeX{}book,} p.\,246} 919 \begin{center} 920 \begin{tabular}{r@{\quad}rl} 921 \hline 922 \multicolumn{1}{l}{\rule{0pt}{12pt} 923 Year}&\multicolumn{2}{l}{World population}\\[2pt] 924 \hline\rule{0pt}{12pt} 925 8000 B.C. & 5,000,000& \\ 926 50 A.D. & 200,000,000& \\ 927 1650 A.D. & 500,000,000& \\ 928 1945 A.D. & 2,300,000,000& \\ 929 1980 A.D. & 4,400,000,000& \\[2pt] 930 \hline 931 \end{tabular} 932 \end{center} 933 \end{table} 934 % 935 \begin{theorem} [Ghoussoub-Preiss] 936 Assume $H(t,x)$ is 937 $(0,\varepsilon )$-subquadratic at 938 infinity for all $\varepsilon > 0$, and $T$-periodic in $t$ 939 \begin{equation} 940 H (t,\cdot )\ \ \ \ \ {\rm is\ convex}\ \ \forall t 941 \end{equation} 942 \begin{equation} 943 H (\cdot ,x)\ \ \ \ \ {\rm is}\ \ T{\rm -periodic}\ \ \forall x 944 \end{equation} 945 \begin{equation} 946 H (t,x)\ge n\left(\left\|x\right\|\right)\ \ \ \ \ 947 {\rm with}\ \ n (s)s^{-1}\to \infty\ \ {\rm as}\ \ s\to \infty 948 \end{equation} 949 \begin{equation} 950 \forall \varepsilon > 0\ ,\ \ \ \exists c\ :\ 951 H(t,x) \le \frac{\varepsilon}{2}\left\|x\right\|^{2} + c\ . 952 \end{equation} 953 954 Assume also that $H$ is $C^{2}$, and $H'' (t,x)$ is positive definite 955 everywhere. Then there is a sequence $x_{k}$, $k\in \bbbn$, of 956 $kT$-periodic solutions of the system 957 \begin{equation} 958 \dot{x} = JH' (t,x) 959 \end{equation} 960 such that, for every $k\in \bbbn$, there is some $p_{o}\in\bbbn$ with: 961 \begin{equation} 962 p\ge p_{o}\Rightarrow x_{pk} \ne x_{k}\ . 963 \end{equation} 964 \qed 965 \end{theorem} 966 % 967 \begin{example} [{{\rm External forcing}}] 968 Consider the system: 969 \begin{equation} 970 \dot{x} = JH' (x) + f(t) 971 \end{equation} 972 where the Hamiltonian $H$ is 973 $\left(0,b_{\infty}\right)$-subquadratic, and the 974 forcing term is a distribution on the circle: 975 \begin{equation} 976 f = \frac{d}{dt} F + f_{o}\ \ \ \ \ 977 {\rm with}\ \ F\in L^{2} \left(\bbbr / T\bbbz; \bbbr^{2n}\right)\ , 978 \end{equation} 979 where $f_{o} : = T^{-1}\int_{o}^{T} f (t) dt$. For instance, 980 \begin{equation} 981 f (t) = \sum_{k\in \bbbn} \delta_{k} \xi\ , 982 \end{equation} 983 where $\delta_{k}$ is the Dirac mass at $t= k$ and 984 $\xi \in \bbbr^{2n}$ is a 985 constant, fits the prescription. This means that the system 986 $\dot{x} = JH' (x)$ is being excited by a 987 series of identical shocks at interval $T$. 988 \end{example} 989 % 990 \begin{definition} 991 Let $A_{\infty} (t)$ and $B_{\infty} (t)$ be symmetric 992 operators in $\bbbr^{2n}$, depending continuously on 993 $t\in [0,T]$, such that 994 $A_{\infty} (t) \le B_{\infty} (t)$ for all $t$. 995 996 A Borelian function 997 $H: [0,T]\times \bbbr^{2n} \to \bbbr$ 998 is called 999 $\left(A_{\infty} ,B_{\infty}\right)$-{\it subquadratic at infinity} 1000 if there exists a function $N(t,x)$ such that: 1001 \begin{equation} 1002 H (t,x) = \frac{1}{2} \left(A_{\infty} (t) x,x\right) + N(t,x) 1003 \end{equation} 1004 \begin{equation} 1005 \forall t\ ,\ \ \ N(t,x)\ \ \ \ \ 1006 {\rm is\ convex\ with\ respect\ to}\ \ x 1007 \end{equation} 1008 \begin{equation} 1009 N(t,x) \ge n\left(\left\|x\right\|\right)\ \ \ \ \ 1010 {\rm with}\ \ n(s)s^{-1}\to +\infty\ \ {\rm as}\ \ s\to +\infty 1011 \end{equation} 1012 \begin{equation} 1013 \exists c\in \bbbr\ :\ \ \ H (t,x) \le 1014 \frac{1}{2} \left(B_{\infty} (t) x,x\right) + c\ \ \ \forall x\ . 1015 \end{equation} 1016 1017 If $A_{\infty} (t) = a_{\infty} I$ and 1018 $B_{\infty} (t) = b_{\infty} I$, with 1019 $a_{\infty} \le b_{\infty} \in \bbbr$, 1020 we shall say that $H$ is 1021 $\left(a_{\infty},b_{\infty}\right)$-subquadratic 1022 at infinity. As an example, the function 1023 $\left\|x\right\|^{\alpha}$, with 1024 $1\le \alpha < 2$, is $(0,\varepsilon )$-subquadratic at infinity 1025 for every $\varepsilon > 0$. Similarly, the Hamiltonian 1026 \begin{equation} 1027 H (t,x) = \frac{1}{2} k \left\|k\right\|^{2} +\left\|x\right\|^{\alpha} 1028 \end{equation} 1029 is $(k,k+\varepsilon )$-subquadratic for every $\varepsilon > 0$. 1030 Note that, if $k<0$, it is not convex. 1031 \end{definition} 1032 % 1033 1034 \paragraph{Notes and Comments.} 1035 The first results on subharmonics were 1036 obtained by Rabinowitz in \cite{2rab}, who showed the existence of 1037 infinitely many subharmonics both in the subquadratic and superquadratic 1038 case, with suitable growth conditions on $H'$. Again the duality 1039 approach enabled Clarke and Ekeland in \cite{2clar:eke:2} to treat the 1040 same problem in the convex-subquadratic case, with growth conditions on 1041 $H$ only. 1042 1043 Recently, Michalek and Tarantello (see Michalek, R., Tarantello, G. 1044 \cite{2mich:tar} and Tarantello, G. \cite{2tar}) have obtained lower 1045 bound on the number of subharmonics of period $kT$, based on symmetry 1046 considerations and on pinching estimates, as in Sect.~5.2 of this 1047 article. 1048 1049 % 1050 % ---- Bibliography ---- 1051 % 1052 \begin{thebibliography}{} 1053 % 1054 \bibitem[1980]{2clar:eke} 1055 Clarke, F., Ekeland, I.: 1056 Nonlinear oscillations and 1057 boundary-value problems for Hamiltonian systems. 1058 Arch. Rat. Mech. Anal. 78, 315--333 (1982) 1059 1060 \bibitem[1981]{2clar:eke:2} 1061 Clarke, F., Ekeland, I.: 1062 Solutions p\'{e}riodiques, du 1063 p\'{e}riode donn\'{e}e, des \'{e}quations hamiltoniennes. 1064 Note CRAS Paris 287, 1013--1015 (1978) 1065 1066 \bibitem[1982]{2mich:tar} 1067 Michalek, R., Tarantello, G.: 1068 Subharmonic solutions with prescribed minimal 1069 period for nonautonomous Hamiltonian systems. 1070 J. Diff. Eq. 72, 28--55 (1988) 1071 1072 \bibitem[1983]{2tar} 1073 Tarantello, G.: 1074 Subharmonic solutions for Hamiltonian 1075 systems via a $\bbbz_{p}$ pseudoindex theory. 1076 Annali di Matematica Pura (to appear) 1077 1078 \bibitem[1985]{2rab} 1079 Rabinowitz, P.: 1080 On subharmonic solutions of a Hamiltonian system. 1081 Comm. Pure Appl. Math. 33, 609--633 (1980) 1082 1083 \end{thebibliography} 1084 \clearpage 1085 \addtocmark[2]{Author Index} % additional numbered TOC entry 1086 \renewcommand{\indexname}{Author Index} 1087 \printindex 1088 \clearpage 1089 \addtocmark[2]{Subject Index} % additional numbered TOC entry 1090 \markboth{Subject Index}{Subject Index} 1091 \renewcommand{\indexname}{Subject Index} 1092 \input{subjidx.ind} 1093 \end{document}