Physics on Manifolds: Proceedings of the International Colloquium in honour of Yvonne Choquet-Bruhat, Paris, June 3–5, 1992

364 Pages·1994·24.76 MB·English

by  A. M. Anile, G. Alí, V. Romano (auth.), M. Flato, R. Kerner, A. Lichnerowicz (eds.)

Physics on Manifolds
MATHEMATICA L PHYSICS STUDIES
Series Editor:
M. FLATO, Universite de Bourgogne ,Dijon, France
VOLUME 15
Physics on Manifolds
Proceeding sof the International Colloquium
in honour of Yvonne Choquet-Bruhat ,Paris June 3-5, 1992
y
edited by
M. Flato
Physique Mathematique,
Universite de Bourgogne ,Dijon, France
R. Keiner
Laboratoire de Gravitation et Cosmologie Relativistes,
UPMC, Paris, France
and
A. Lichnerowicz
College de France ,Paris, France
SPRINGER SCIENCE+BUSINESS MEDIA , B.V.
Library of Congress Cataloging-in-Publicatino Data
Physics on manifold s : proceedings of th e Internationa l colloquiu m i n
honour of Yvonne Choquet-Bruhat, Paris , June 3-5, 1992 / edite d by
M. Flato , R. Kerner, and A. Uchnerowicz.
p. cm. — (Mathematical physic s studie s ; v. 15)
Eng 1ish and French.
Includes bibliographica l references .
ISBN 978-94-010-4857-6 ISBN 978-94-011-1938-2 (eBook)
DOI 10.1007/978-94-011-1938-2
1. Man1folds(Mathematics) 2. Mathematical physics . I .C hoquet
-Bruhat, Yvonne. II . Flato , M. (Moshe), 1937- . III . Kerner, R.
IV. Lichnerowicz , Andre, 1915- . V. Series .
QC20.7.M24P84 1993
530. V 515~dc20 93-31133
Printed on acid-free paper
All Rights Reservde
© 1994 Springer Science+Busines sMedia Dordrecht
Originally published by Kluwer Academci Publishesr in 1994
Softcover reprint of the hardcover 1st edition 1994
No part of the material protected by this copyright notice may be reproducde or
utilized in any form or by any mean,s electronci or mechanica, l
including photocopying, recording or by any information storage and
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TA BLE DES MA TIERES
Avant-Propos vii
Foreword
IX
I.C. LEGRAND, Discours d'ouverture ................................................................. xv
Conferences plenieres
A.M. ANILE, G. ALi, V. ROMANO, Relativistic dissipative fluids
H. BREZIS, Mathematical problems related to liquid crystals, superconductors and
superfluidr ............................................................................................ II
J.D. BROWN, J.W.YORK Jr., Microcanonical action and the entropy ofa rotating black hole 23
a
F. CAGNAC, M. DOSSA, Probteme de Cauchy sur un canoide caracteristique. Applications
certains systemes non lineaires d'origine physique ............................................... 35
D. CHRISTODOULOU, Recent progress on the Cauchy problem in general relativity ......... 49
T. DAMOUR, On some links between mathematical physics and physics in the context of
general relativity ..................................................................................... 59
C. DEWITT-MORETIE, Functional integration. A multipurpose tool ............................ 67
G. FERRARESE, C. CATIANI, Generalizedframes of references and intrinsic Cauchy
problem in general relativity ........................................................................ 93
A.E. FISCHER, V. MONCRIEF, Reducing Einstein's equations to an unconstrained
Hamiltonian system on the cotangent bundle of Teichmiiller space .. .......... .......... ...... III
C.H. au, Darboux transformations for a class of integrable systems in n variables ............. 153
N.H. IBRAGIMOV, Group theoretical treatment off undamental solutions ........................ 161
S. KLAINERMAN, M. MACHEDON, On the regularity properties of the wave equation. ...... 177
J. LERA Y, Le probleme de Cauchy lineaire et analytique pour un opirateur holomorphe et un
second membre ram (fie [Resume) ................................................................. 193
P.L. LIONS, On Boltzmann equation .................................................................. 195
C. MORENO, L. VALERO, Star products and quantum groups ................................... 203
O. A. OLEINIK, On asymptotic of solutions of a nonlinear elliptic equation
in a cylindrical domai:J ........................................................ . 235
I. SEGAL, Fundamental physics in universal space-time ............................................ 253
A.H. TAUB, Interaction ofg ravitational and electromagnetic waves in general relativity ........ 265
C. TAUBES, Anti-self dual conformal structures on4-m((nifolds [Resume) ....................... 289
vi TABLE DES MATIERES
Tables ron des
Equations differentielles aux derivees partielles (presidie par J. VAILLANT)
E. CALZETfA, Chaotic behavior in relativistic motion ............................................... 291
J. ISENB~RG~ V. MO~CRIEF, .Some results on non constant mean curvature solutions of
the Einstein constraint equatIOns .................................................................... 295
W. MATSUMOTO, Levi condition for general systems .............................................. 303
J. VAILLANT, Conditions invariantes pour un systeme, du type conditions de Levi ............. 309
Physique MatMmatique (presidee par G. PICHON)
P.C. AICHELBURG, Black holes in supergravity ................ ..... ............. ........ ....... ... 315
E.A. CHRISTENSEN, J.N. SORENSEN, M. BRONS, P.L. CHRISTIANSEN,
Low-dimensional behaviour in the rotating driven cavity problem ............... .............. 321
M. EPSTEIN, G. A. MAUGIN, Some geometrical aspects of inhomogeneous elasticity ........ 331
T. BRUGARINO, A. M. GRECO, Integrating the Kadomtsev-Petviashvili equation in the
1+ 3 dimensions via the generalised Monge-Ampere equation: an example of conditioned
Pa inleve test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
V.S. MANKO, N.R. SIBGATULLIN, Spinning mass endowed with electric charge and
magnetic dipole moment .. .............................. .................. ....... ....... ............. 347
G. PICHON, Equations de Vlasoven tMorie discrete ................................................ 353
T. RUGGERI, Convexity and symmetrization in classical and relativistic balance laws systelns 357
AVANT·PROPOS
Ce volume contient les actes du Colloque "Analyse, Varietes et Physique", organise en I'honneur
d'Yvonne Choquet-Bruhat, les 3, 4 et 5 juin 1992 a Paris, par ses amis, collaborateurs et anciens eleves.
Son titre reflete assez bien les domaines d'activite dans lesquels Yvonne Choquet-Bruhat apporta des
contributions essentielles.
Depuis I'avenement de la Relativite Generale, la geometrie des varietes est devenue une partie non
triviale de la physique de I'espace-temps. L'analyse fonctionnelle a trouve en meme temps une
application essentielle en mecanique quantique, puis en theorie quantique des champs. Son apport
devient decisif au moment ou I'on considere Ie comportement global des solutions des systemes
differentiels sur les varietes. La Relativite Generale est dans ce sens une theorie exceptionnelle, dans
laquelle les solutions d'un systeme hautement non lineaire d'equations aux derivees partielles definissent
elles-memes la variete sur laquelle elles sont supposees exister. C'est pOUl'quoi aucune des solutions des
equations d'Einstein ne peut etre interpl-etee physiquement avant que son comportement global, c'est-a
dire sur I'ensemble de la variete hypothetique, ne soit connu.
C'est a ce domaine, se situ ant au carrefour entre la physique et les mathematiques, que Yvonne
Choquet-Bruhat a contribue, de maniere spectaculaire dans sa jeunesse, en donnant la preuve de
l'existence des solutions des equations d'Einstein sur des varietes differentiables assez generales. Les
methodes qu'elle avait creees ont ete elaborees par l'ecole fran<;aise, principalement par Jean Leray. Cette
premiere preuve de I'existence et de I'unicite locale pour les equations d'Einstein a ete Ie moteur de la
theorie des systemes hyperboliques generaux de Jean Leray.
Yvonne Choquet-Bruhat a repandu dans la communaute scientifique l'idee de la necessite de
l'etude des problemes globaux imposes par la physique de systemes couples a la gravitation. En ce qui
concerne Ie systeme d'equations d'Einstein, Ie but a ete atteint tres recemment par les travaux de
S. Klainerman et de D. ChIistodoulou, ancien collaborateur d'Yvonne Choquet-Bruhat.
Face au developpement de la physique contemporaine, Yvonne Choquet-Bruhat a toujours joue un
rOle de pionnier dans l'etude de problemes globaux, notamment dans la themie des champs de jauge et la
supergravite. Parallelement, elle a construit une theorie des ondes asymptotiques gravitationnelles et
electromagnetiques. Son influence s'est aussi fait sentir par l'elaboration de livres de base, devenus
classiques, consacres a l'analyse fonctionnelle sur les vaIietes et a ses applications dans differents
domaines de la physique.
Yvonne Choquet-Bruhat s'est aussi devouee a la formation de nombreux jeunes chercheurs
fran<;ais et etrangers, dont certains sont devenus des maitres.
Sa modestie n'a jamais occuIte ses qualites humaines exceptionnelles et son sens profond de
l'amitie ",nvers ses collaborateurs, eleves et amis.
Nous dedions ce livre, qui donne la mesure de son rayonnement dans la communaute scientifique,
a Yvonne Choquet-Bruhat, en souhaitant qu'il soit digne de sa personne et de son oeuvre.
M. Flato, R. Kerner, A. Lichnerowicz
vii
FOREWORD
This volume contains the proceedings of the Colloquium "Analysis, Manifolds and Physics"
organized in honour of Yvonne Choquet-Bruhat by her friends, collaborators and former students, on
June 3, 4 and 5, 1992 in Paris. Its title accurately reflects the domains to which Yvonne Choquet-Bruhat
has made essential contributions.
Since the rise of General Relativity, the geometry of Manifolds has become a non-trivial part of
space-time physics. At the same time, Functional Analysis has been of enormous importance in
Quantum Mechanics, and Quantum Field Theory. Its role becomes decisive when one considers the
global behaviour of solutions of differential systems on manifolds. In this sense, General Relativity is an
exceptional theory in which the solutions of a highly non-linear system of partial differential equations
define by themselves the very manifold on which they are supposed to exist. This is why a solution of
Einstein's equations cannot be physically interpreted before its global behaviour is known, taking into
account the entire hypothetical underlying manifold.
In her youth, Yvonne Choquet-Bruhat contributed in a spectacular way to this domain stretching
between physics and mathematics, when she gave the proof of the existence of solutions to Einstein's
equations on differential manifolds of a quite general type. The methods she created have been worked
out by the French school of mathematics, principally by Jean Leray. Her first proof of the local existence
and uniqueness of solutions of Einstein's equations inspired Jean Leray's theory of general hyperbolic
systems.
Yvonne Choquet-Bruhat has promoted in the scientific community the idea of the necessity of
studying the global problems posed by the physics of systems coupled with gravity. A complete study
of global properties of solutions to Einstein's equations has been achieved recently by S. Klainerman
and by her former collaborator D. Chlistodoulou.
With regard to recent development of contemporary physics, Yvonne Choquet-Bruhat has played a
pioneeling r6\e in the study of global problems, especially in gauge field theory and supergravity. At the
same time, she has elaborated a theory of asymptotic gravitational and electromagnetic waves. Her
influence has also been felt through new basic books, which have since become classics, concerning
functional analysis on manifolds and its applications in various domains of physics. Yvonne Choquet
Bruhat has also devoted herself to the development of many young research students in France and
abroad, some of whom have become masters.
Her modesty has never eclipsed her truly exceptional human qualities and her profound sense of
fliendship towards her collaborators, students and fIiends.
We dedicate this book, which illustrates the full measure of her influence in the scientific
community, to Yvonne Choquet-Bruhat, and we hope that it will be worthy of her and her work.
M. Flato, R. Kerner, A. Lichnerowicz
ix
COMITE D'HONNEUR
Bryce DEWm (Austin, USA) Andre LICHNEROWICZ (Paris, France)
Israel GELFAND (Piscataway, USA) Jacques-Louis LIONS (Paris, France)
Paul GERMAIN (Paris, France) Irving E. SEGAL (Cambridge, USA)
Jean LERA Y (Paris, France) Abraham H. TAUB (Berkeley, USA)
COMITE D'ORGANISA nON
Daniel BANCEL (Lyon, France) Andre LICHNEROWICZ (Paris, France)
Moshe FLATO (Dijon, France) Charles-Michel MARLE (Paris, France)
Krzysztof GA WEDZKI (B ures S/Yvette, France) Guy PICHON (Villetaneuse, France)
Richard KERNER (Paris, France) Jean VAILLANT (Paris, France)
Lise LAMOUREUX-BROUSSE (Paris, France)
LISTE DES CONFERENCIERS
(conferences plenieres et tables rondes)
Peter C. AICHELBURG (Vienne, Autriche) Jean LERA Y (Paris, France)
Angelo M. ANILE (Catane, Italie) Pierre-Louis LIONS (Paris, France)
Haim BREZIS (Paris, France) Waichiro MATSUMOTO (Ohtsu, Japon)
Francis CAGNAC (Yaounde, Cameroun) Gerard A. MAUGIN (Paris, France)
Esteban CALZETIA (Buenos Aires, Argentine) Vincent MONCRIEF (New Haven, USA)
Erik A. CHRISTENSEN (Lyngby, Danemark) Carlos MORENO (Madrid, Espagne)
Demetrios CHRISTODOULOU (Princeton, USA) Olga A. OLEINIK (Moscou, Russie)
Thibault DAMOUR (Bures-sur-Yvette, France) Guy PICHON (Villetaneuse, France)
Cecile DEWTIT-MORETTE (Austin, USA) Tommaso RUGGERI (Bologne, Italie)
Giorgio FERRARESE (Rome, Italie) Irving E. SEGAL (Cambridge, USA)
Antonio GRECO (Palerme, Italie) Nail R. SIBGATULLIN (Moscou, Russie)
ChaoHao GU (Hefei, Rep. Pop. de Chine) Abraham H. TAUB (Berkeley, USA)
Nail H. IBRAGIMOV (Moscou, Russie) Clifford TAUBES (Cambridge, USA)
Jim ISENBERG (Eugene, USA) Jean VAILLANT (Paris, France)
Sergiu KLAINERMAN (Princeton, USA) James W. YORK Jr. (Chapel Hill, USA)