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\documentclass{llncs}
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\usepackage{makeidx} % allows for indexgeneration
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\begin{document}
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\frontmatter % for the preliminaries
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%
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\pagestyle{headings} % switches on printing of running heads
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\addtocmark{Hamiltonian Mechanics} % additional mark in the TOC
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%
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\chapter*{Preface}
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%
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This textbook is intended for use by students of physics, physical
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chemistry, and theoretical chemistry. The reader is presumed to have a
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basic knowledge of atomic and quantum physics at the level provided, for
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example, by the first few chapters in our book {\it The Physics of Atoms
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and Quanta}. The student of physics will find here material which should
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be included in the basic education of every physicist. This book should
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furthermore allow students to acquire an appreciation of the breadth and
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variety within the field of molecular physics and its future as a
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fascinating area of research.
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For the student of chemistry, the concepts introduced in this book will
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provide a theoretical framework for that entire field of study. With the
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help of these concepts, it is at least in principle possible to reduce
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the enormous body of empirical chemical knowledge to a few basic
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principles: those of quantum mechanics. In addition, modern physical
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methods whose fundamentals are introduced here are becoming increasingly
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important in chemistry and now represent indispensable tools for the
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chemist. As examples, we might mention the structural analysis of
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complex organic compounds, spectroscopic investigation of very rapid
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reaction processes or, as a practical application, the remote detection
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of pollutants in the air.
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\vspace{1cm}
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\begin{flushright}\noindent
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April 1995\hfill Walter Olthoff\\
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Program Chair\\
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ECOOP'95
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\end{flushright}
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%
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\chapter*{Organization}
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ECOOP'95 is organized by the department of Computer Science, Univeristy
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of \AA rhus and AITO (association Internationa pour les Technologie
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Object) in cooperation with ACM/SIGPLAN.
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%
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\section*{Executive Commitee}
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\begin{tabular}{@{}p{5cm}@{}p{7.2cm}@{}}
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Conference Chair:&Ole Lehrmann Madsen (\AA rhus University, DK)\\
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Program Chair: &Walter Olthoff (DFKI GmbH, Germany)\\
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Organizing Chair:&J\o rgen Lindskov Knudsen (\AA rhus University, DK)\\
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Tutorials:&Birger M\o ller-Pedersen\hfil\break
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(Norwegian Computing Center, Norway)\\
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Workshops:&Eric Jul (University of Kopenhagen, Denmark)\\
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Panels:&Boris Magnusson (Lund University, Sweden)\\
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Exhibition:&Elmer Sandvad (\AA rhus University, DK)\\
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Demonstrations:&Kurt N\o rdmark (\AA rhus University, DK)
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\end{tabular}
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%
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\section*{Program Commitee}
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\begin{tabular}{@{}p{5cm}@{}p{7.2cm}@{}}
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Conference Chair:&Ole Lehrmann Madsen (\AA rhus University, DK)\\
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Program Chair: &Walter Olthoff (DFKI GmbH, Germany)\\
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Organizing Chair:&J\o rgen Lindskov Knudsen (\AA rhus University, DK)\\
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|
Tutorials:&Birger M\o ller-Pedersen\hfil\break
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(Norwegian Computing Center, Norway)\\
|
|
Workshops:&Eric Jul (University of Kopenhagen, Denmark)\\
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|
Panels:&Boris Magnusson (Lund University, Sweden)\\
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Exhibition:&Elmer Sandvad (\AA rhus University, DK)\\
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Demonstrations:&Kurt N\o rdmark (\AA rhus University, DK)
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\end{tabular}
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%
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\begin{multicols}{3}[\section*{Referees}]
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V.~Andreev\\
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B\"arwolff\\
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E.~Barrelet\\
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H.P.~Beck\\
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G.~Bernardi\\
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E.~Binder\\
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P.C.~Bosetti\\
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Braunschweig\\
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F.W.~B\"usser\\
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T.~Carli\\
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A.B.~Clegg\\
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G.~Cozzika\\
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S.~Dagoret\\
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Del~Buono\\
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P.~Dingus\\
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H.~Duhm\\
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J.~Ebert\\
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S.~Eichenberger\\
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R.J.~Ellison\\
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Feltesse\\
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W.~Flauger\\
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A.~Fomenko\\
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G.~Franke\\
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J.~Garvey\\
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M.~Gennis\\
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L.~Goerlich\\
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P.~Goritchev\\
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H.~Greif\\
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E.M.~Hanlon\\
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R.~Haydar\\
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R.C.W.~Henderso\\
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P.~Hill\\
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H.~Hufnagel\\
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A.~Jacholkowska\\
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Johannsen\\
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S.~Kasarian\\
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I.R.~Kenyon\\
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C.~Kleinwort\\
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T.~K\"ohler\\
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S.D.~Kolya\\
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P.~Kostka\\
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U.~Kr\"uger\\
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J.~Kurzh\"ofer\\
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M.P.J.~Landon\\
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A.~Lebedev\\
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Ch.~Ley\\
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F.~Linsel\\
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H.~Lohmand\\
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Martin\\
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S.~Masson\\
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K.~Meier\\
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C.A.~Meyer\\
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S.~Mikocki\\
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J.V.~Morris\\
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B.~Naroska\\
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Nguyen\\
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U.~Obrock\\
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G.D.~Patel\\
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Ch.~Pichler\\
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S.~Prell\\
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F.~Raupach\\
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V.~Riech\\
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P.~Robmann\\
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N.~Sahlmann\\
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P.~Schleper\\
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Sch\"oning\\
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B.~Schwab\\
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A.~Semenov\\
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G.~Siegmon\\
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J.R.~Smith\\
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M.~Steenbock\\
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U.~Straumann\\
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C.~Thiebaux\\
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P.~Van~Esch\\
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from Yerevan Ph\\
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L.R.~West\\
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G.-G.~Winter\\
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T.P.~Yiou\\
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M.~Zimmer\end{multicols}
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%
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\section*{Sponsoring Institutions}
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%
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Bernauer-Budiman Inc., Reading, Mass.\\
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The Hofmann-International Company, San Louis Obispo, Cal.\\
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Kramer Industries, Heidelberg, Germany
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%
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\tableofcontents
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%
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\mainmatter % start of the contributions
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%
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\title{Hamiltonian Mechanics unter besonderer Ber\"ucksichtigung der
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h\"ohreren Lehranstalten}
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%
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\titlerunning{Hamiltonian Mechanics} % abbreviated title (for running head)
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% also used for the TOC unless
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% \toctitle is used
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%
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\author{Ivar Ekeland\inst{1} \and Roger Temam\inst{2}
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Jeffrey Dean \and David Grove \and Craig Chambers \and Kim~B.~Bruce \and
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Elsa Bertino}
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%
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\authorrunning{Ivar Ekeland et al.} % abbreviated author list (for running head)
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%
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%%%% modified list of authors for the TOC (add the affiliations)
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\tocauthor{Ivar Ekeland (Princeton University),
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Roger Temam (Universit\'{e} de Paris-Sud),
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Jeffrey Dean, David Grove, Craig Chambers (Universit\`a di Geova),
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Kim B. Bruce (Stanford University),
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Elisa Bertino (Digita Research Center)}
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%
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\institute{Princeton University, Princeton NJ 08544, USA,\\
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\email{I.Ekeland@princeton.edu},\\ WWW home page:
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\texttt{http://users/\homedir iekeland/web/welcome.html}
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\and
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Universit\'{e} de Paris-Sud,
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Laboratoire d'Analyse Num\'{e}rique, B\^{a}timent 425,\\
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F-91405 Orsay Cedex, France}
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\maketitle % typeset the title of the contribution
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\begin{abstract}
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The abstract should summarize the contents of the paper
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using at least 70 and at most 150 words. It will be set in 9-point
|
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font size and be inset 1.0 cm from the right and left margins.
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There will be two blank lines before and after the Abstract. \dots
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\end{abstract}
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%
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\section{Fixed-Period Problems: The Sublinear Case}
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%
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With this chapter, the preliminaries are over, and we begin the search
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for periodic solutions to Hamiltonian systems. All this will be done in
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the convex case; that is, we shall study the boundary-value problem
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\begin{eqnarray*}
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\dot{x}&=&JH' (t,x)\\
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x(0) &=& x(T)
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\end{eqnarray*}
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with $H(t,\cdot)$ a convex function of $x$, going to $+\infty$ when
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$\left\|x\right\| \to \infty$.
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%
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\subsection{Autonomous Systems}
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%
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In this section, we will consider the case when the Hamiltonian $H(x)$
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is autonomous. For the sake of simplicity, we shall also assume that it
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is $C^{1}$.
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We shall first consider the question of nontriviality, within the
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general framework of
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$\left(A_{\infty},B_{\infty}\right)$-subquadratic Hamiltonians. In
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the second subsection, we shall look into the special case when $H$ is
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$\left(0,b_{\infty}\right)$-subquadratic,
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and we shall try to derive additional information.
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%
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\subsubsection{The General Case: Nontriviality.}
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%
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We assume that $H$ is
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$\left(A_{\infty},B_{\infty}\right)$-sub\-qua\-dra\-tic at infinity,
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for some constant symmetric matrices $A_{\infty}$ and $B_{\infty}$,
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with $B_{\infty}-A_{\infty}$ positive definite. Set:
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\begin{eqnarray}
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\gamma :&=&{\rm smallest\ eigenvalue\ of}\ \ B_{\infty} - A_{\infty} \\
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\lambda : &=& {\rm largest\ negative\ eigenvalue\ of}\ \
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J \frac{d}{dt} +A_{\infty}\ .
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\end{eqnarray}
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Theorem~\ref{ghou:pre} tells us that if $\lambda +\gamma < 0$, the
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boundary-value problem:
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\begin{equation}
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\begin{array}{rcl}
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\dot{x}&=&JH' (x)\\
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x(0)&=&x (T)
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\end{array}
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\end{equation}
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has at least one solution
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$\overline{x}$, which is found by minimizing the dual
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action functional:
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\begin{equation}
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\psi (u) = \int_{o}^{T} \left[\frac{1}{2}
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\left(\Lambda_{o}^{-1} u,u\right) + N^{\ast} (-u)\right] dt
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\end{equation}
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on the range of $\Lambda$, which is a subspace $R (\Lambda)_{L}^{2}$
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with finite codimension. Here
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\begin{equation}
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N(x) := H(x) - \frac{1}{2} \left(A_{\infty} x,x\right)
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\end{equation}
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is a convex function, and
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\begin{equation}
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N(x) \le \frac{1}{2}
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\left(\left(B_{\infty} - A_{\infty}\right) x,x\right)
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+ c\ \ \ \forall x\ .
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\end{equation}
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%
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\begin{proposition}
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Assume $H'(0)=0$ and $ H(0)=0$. Set:
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\begin{equation}
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\delta := \liminf_{x\to 0} 2 N (x) \left\|x\right\|^{-2}\ .
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\label{eq:one}
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\end{equation}
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If $\gamma < - \lambda < \delta$,
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the solution $\overline{u}$ is non-zero:
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\begin{equation}
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\overline{x} (t) \ne 0\ \ \ \forall t\ .
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\end{equation}
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\end{proposition}
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%
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\begin{proof}
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Condition (\ref{eq:one}) means that, for every
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$\delta ' > \delta$, there is some $\varepsilon > 0$ such that
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\begin{equation}
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\left\|x\right\| \le \varepsilon \Rightarrow N (x) \le
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\frac{\delta '}{2} \left\|x\right\|^{2}\ .
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\end{equation}
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It is an exercise in convex analysis, into which we shall not go, to
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show that this implies that there is an $\eta > 0$ such that
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\begin{equation}
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f\left\|x\right\| \le \eta
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\Rightarrow N^{\ast} (y) \le \frac{1}{2\delta '}
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\left\|y\right\|^{2}\ .
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\label{eq:two}
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\end{equation}
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\begin{figure}
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\vspace{2.5cm}
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\caption{This is the caption of the figure displaying a white eagle and
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a white horse on a snow field}
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\end{figure}
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Since $u_{1}$ is a smooth function, we will have
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$\left\|hu_{1}\right\|_\infty \le \eta$
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for $h$ small enough, and inequality (\ref{eq:two}) will hold,
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yielding thereby:
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\begin{equation}
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\psi (hu_{1}) \le \frac{h^{2}}{2}
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\frac{1}{\lambda} \left\|u_{1} \right\|_{2}^{2} + \frac{h^{2}}{2}
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\frac{1}{\delta '} \left\|u_{1}\right\|^{2}\ .
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\end{equation}
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If we choose $\delta '$ close enough to $\delta$, the quantity
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$\left(\frac{1}{\lambda} + \frac{1}{\delta '}\right)$
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will be negative, and we end up with
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\begin{equation}
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\psi (hu_{1}) < 0\ \ \ \ \ {\rm for}\ \ h\ne 0\ \ {\rm small}\ .
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\end{equation}
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On the other hand, we check directly that $\psi (0) = 0$. This shows
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that 0 cannot be a minimizer of $\psi$, not even a local one.
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So $\overline{u} \ne 0$ and
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$\overline{u} \ne \Lambda_{o}^{-1} (0) = 0$. \qed
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\end{proof}
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%
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\begin{corollary}
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Assume $H$ is $C^{2}$ and
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$\left(a_{\infty},b_{\infty}\right)$-subquadratic at infinity. Let
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$\xi_{1},\allowbreak\dots,\allowbreak\xi_{N}$ be the
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equilibria, that is, the solutions of $H' (\xi ) = 0$.
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Denote by $\omega_{k}$
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the smallest eigenvalue of $H'' \left(\xi_{k}\right)$, and set:
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\begin{equation}
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\omega : = {\rm Min\,} \left\{\omega_{1},\dots,\omega_{k}\right\}\ .
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\end{equation}
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If:
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\begin{equation}
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\frac{T}{2\pi} b_{\infty} <
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- E \left[- \frac{T}{2\pi}a_{\infty}\right] <
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\frac{T}{2\pi}\omega
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\label{eq:three}
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\end{equation}
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then minimization of $\psi$ yields a non-constant $T$-periodic solution
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$\overline{x}$.
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\end{corollary}
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%
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We recall once more that by the integer part $E [\alpha ]$ of
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$\alpha \in \bbbr$, we mean the $a\in \bbbz$
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such that $a< \alpha \le a+1$. For instance,
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if we take $a_{\infty} = 0$, Corollary 2 tells
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us that $\overline{x}$ exists and is
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non-constant provided that:
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\begin{equation}
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\frac{T}{2\pi} b_{\infty} < 1 < \frac{T}{2\pi}
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\end{equation}
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or
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|
\begin{equation}
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T\in \left(\frac{2\pi}{\omega},\frac{2\pi}{b_{\infty}}\right)\ .
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\label{eq:four}
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\end{equation}
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%
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\begin{proof}
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The spectrum of $\Lambda$ is $\frac{2\pi}{T} \bbbz +a_{\infty}$. The
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largest negative eigenvalue $\lambda$ is given by
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$\frac{2\pi}{T}k_{o} +a_{\infty}$,
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|
where
|
|
\begin{equation}
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\frac{2\pi}{T}k_{o} + a_{\infty} < 0
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\le \frac{2\pi}{T} (k_{o} +1) + a_{\infty}\ .
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\end{equation}
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|
Hence:
|
|
\begin{equation}
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|
k_{o} = E \left[- \frac{T}{2\pi} a_{\infty}\right] \ .
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|
\end{equation}
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|
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|
The condition $\gamma < -\lambda < \delta$ now becomes:
|
|
\begin{equation}
|
|
b_{\infty} - a_{\infty} <
|
|
- \frac{2\pi}{T} k_{o} -a_{\infty} < \omega -a_{\infty}
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|
\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{clar: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{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{rab}, 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{clar:eke:2} to treat the
|
|
same problem in the convex-subquadratic case, with growth conditions on
|
|
$H$ only.
|
|
|
|
Recently, Michalek and Tarantello (see \cite{mich:tar} and \cite{tar})
|
|
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}{5}
|
|
%
|
|
\bibitem {clar:eke}
|
|
Clarke, F., Ekeland, I.:
|
|
Nonlinear oscillations and
|
|
boundary-value problems for Hamiltonian systems.
|
|
Arch. Rat. Mech. Anal. {\bf 78} (1982) 315--333
|
|
|
|
\bibitem {clar:eke:2}
|
|
Clarke, F., Ekeland, I.:
|
|
Solutions p\'{e}riodiques, du
|
|
p\'{e}riode donn\'{e}e, des \'{e}quations hamiltoniennes.
|
|
Note CRAS Paris {\bf 287} (1978) 1013--1015
|
|
|
|
\bibitem {mich:tar}
|
|
Michalek, R., Tarantello, G.:
|
|
Subharmonic solutions with prescribed minimal
|
|
period for nonautonomous Hamiltonian systems.
|
|
J. Diff. Eq. {\bf 72} (1988) 28--55
|
|
|
|
\bibitem {tar}
|
|
Tarantello, G.:
|
|
Subharmonic solutions for Hamiltonian
|
|
systems via a $\bbbz_{p}$ pseudoindex theory.
|
|
Annali di Matematica Pura (to appear)
|
|
|
|
\bibitem {rab}
|
|
Rabinowitz, P.:
|
|
On subharmonic solutions of a Hamiltonian system.
|
|
Comm. Pure Appl. Math. {\bf 33} (1980) 609--633
|
|
|
|
\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
|
|
\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. {\bf 78} (1982) 315--333
|
|
|
|
\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 {\bf 287} (1978) 1013--1015
|
|
|
|
\bibitem[1982]{2mich:tar}
|
|
Michalek, R., Tarantello, G.:
|
|
Subharmonic solutions with prescribed minimal
|
|
period for nonautonomous Hamiltonian systems.
|
|
J. Diff. Eq. {\bf 72} (1988) 28--55
|
|
|
|
\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. {\bf 33} (1980) 609--633
|
|
|
|
\end{thebibliography}
|
|
\clearpage
|
|
\addtocmark[2]{Author Index} % additional numbered TOC entry
|
|
\renewcommand{\indexname}{Author Index}
|
|
\printindex
|
|
\clearpage
|
|
\addtocmark[2]{Subject Index} % additional numbered TOC entry
|
|
\markboth{Subject Index}{Subject Index}
|
|
\renewcommand{\indexname}{Subject Index}
|
|
\input{subjidx.ind}
|
|
\end{document}
|