Table Of ContentLecture Notes ni
Mathematics
Edited by .A Dold dna .B nnamkcE
775
Geometric Methods ni
Mathematical Physics
Proceedings of na NSF-CBMS Conference
Held ta the University of Lowell,
Massachusetts, March 19-23, 1979
Edited by
.G Kaiser . E.and J Marsden
galreV-regnirpS
Berlin Heidelberg New York 1980
Editors
Gerald Kaiser
Mathematics Department
University of Lowell
Lowell, MA 01854
USA
Jerrold E. Marsden
Department of Mathematics
University of California
Berkeley, CA 94720
USA
AMS Subject Classifications (1980): 53CXX, 58FXX, 73C50, 81-XX,
83CXX
ISBN 3-540-09742-2 Springer-Verlag Berlin Heidelberg NewYork
ISBN 0-38?-09742-2 Springer-Verlag NewYork Heidelberg Berlin
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Introduction
This volume represents invited papers presented at the CBMS regional
conference held at the University of Lowell~ March 19-23. The theme of the con-
ference was geometric methods in mathematical physics and the papers were chosen
with this in mind.
It is really only in the last couple of decades that the usefulness of
geometric methods in mathematical physics has been brought to light. In other
branches of mathematics their usefulness has been clearly demonstrated by Riemann,
Poincare and Cartan; a modern example is the use of symp!ectic geometry in group
representations by Kirillov and Kostant. Save for general relativity~ mathematical
physics has been dominated primarily by analytical techniques. The excitement of
the past few decades has been the complementing power of geometric methods.
The proper geometrization of classical mechanics started with Poincare and con-
tinued with many workers, such as Synge (Phil. Trans. (1926)) and Reeb .C( R. Acad.
Sci. (1948)). However, it wasn't until the analysis led to and became inextricably
involved with geometry through the deep works of Kolmogorov, Arnold and Moser in
celestical mechanics that a permanent bond became reality. The success of
symplectic geometry in classical mechanics has motivated attempts to extend its use
to the quantum domain. Some of these have borne rich fruit, such as the discovery
of the geometry behind the WKB approximation (semiclassical mechanics) by Keller
and Maslov and the quantization program of Souriau and Kostant. SyTaplectic
geometry and classical mechanics have also revitalized linear partial differential
equations through the work of Egorov, HSrmander, Nirenberg and Treves.
Much work is currently going on in gauge theory and supersymmetry that is, of
necessity, geometric. Some believe that these geometric methods will finally close
the circle with relativity as Einstein had dreamed.
.G Kaiser
J. Marsden
April, 1979
Acknowle.dgements
I wish to thank the following University of Lowell faculty members for their
help in organizing this exciting conference: Alan Doerr, Lloyd Kannenberg,
Eric Sheldon and Virginia Taylor. I am also grateful to the National Science
Foundation and the Conference Board of the Mathematical Sciences for sponsoring
the conference, and to Jerry Marsden for lighting the fire.
Gerald Kaiser
TABLE OF CONTENTS
S. Antman: GEOMETRIC ASPECTS OF GLOBAL
BIFURCATION IN NONLINEAR ELASTICITY ....
V. Moncrief: THE BRANCHING OF SOLUTIONS
OF EINSTEIN'S EQUATIONS . . . . . . . . . . 30
S. Deser: WhAT DOES SUPERGRAVITY TEACH
US ABOUT GFAVITY? . . . . . . . . . . . . . h9
C. Galvao: CLASSICAL ½-SPIN PARTICLES
WITH GRAVITATIONAL FIELDS: A SUPER-
SYMMETRIC MODEL . . . . . . . . . . . . . . 69
M. Gotay and J. Nester: GENERALIZED
CONSTRAINT ALGORITHM AND SPECIAL PRE-
SYMPLECTIC MANIFOLDS . . . . . . . . . . . 78
A. Lichnerowicz: DEFORMATIONS ANN
QUANTIZATION . . . . . . . . . . . . . . . ~o5
G. Kaiser: HOLOMORPHIC GAUGE THECRY . . . . . 122
R. Hermann: A GEOMETRIC VARIATIONAL
FORMALISM FOR THE THEORY OF
NONLINEAR WAVES . . . . . . . . . . . . . 145
B. Kupershmidt: GEOMETRY OF JET BUNDLES
AND THE STRUCTURE OF LAGRANGIAN AND
HAMILTONIAN FORMALISMS . . . . . . . . . . 162
T. Ratiu: INVOLUTION THEOREMS . . . . . . . . 219
List of Participants
Demis Aebersold, Physics & Chemistr~y Dept., Bennington College, Bennington, VT
St~rrt Antman, Div. of Applied Math, Box F, Brown Univ., Providence, RI
Timothy Bock, 115 Broadmead, Princeton• NJ
.R Bolger, Fairfield Univ., Fairfield, CT
Bohumil Cenkl, ~th Dept. • Northeastern Univ., Boston, &l~
William Crombie, P.O. Box 7025, Brown Univ., Providence, RI
Richard Cushman, Mathematics - Natural Science II, U of California, Santa Cruz, CA
Stanley Deser, Physics Dept., Brandeis U., Waltham, MA
Robert Devaney, Math Dept., Tufts Univ., Medford, MA
Alan Doerr, Math Dept., U. of Lowell, Lowell, K&
Alexander Doohovsky, 63 Brooks St., Concord, HA
Gerard (cid:127)meh, }~th Dept., .U of Rochester• Rochester• }~
Carlos GalvZo, Physics Dept., Princeton ~ Univ. Princeton, NJ
.P .L Garcia, Math Dept., .vi~YU of Salamanca,~ Salamanca, Spain
Maurice Gilmore, Math Dept., Northeastern Univ., Boston, MA
Daniel Goroff, Churchill College, Cambridge U., Cambridge, England
M~k @otay, Physics Dept., U. of Maryland, College Park, MD
Morton Gt<rtin, Math Dept., Carnegie-Mellon U., Pittsburgh, PA
.S H~a~iharan, Math Dept., Carnegie-Mellon U., Pittsburgh, PA
.H Hattori, Math Dept., Rensselaer Polytechnic Institute, Troy, NY
Robert Herm~r~n, Math-Sci Press• 53 Jordan Rd., Brookline, ~'[
Gerald Kaiser, Math Dept., .U of Lowell, Lowell, MA
Lloyd Kannenberg, Physics Dept., U. of Lowell, Lowell, MA
Boris Kupershmidt, Math Dept., M.I.T., Cambridge, MA
Henry Kurland, Math Dept., Boston [niv., Boston, MA
VII'
Andr~ Lichnerowicz, M.I.T. and College de France, Paris
Jerrold Marsden, Math Dept., U. of California - Berkeley, Berkeley, CA
Vincent Moncrief, Physics Dept., Yale Univ. ,New Haven, CT
Harry Moses, .U of Lowell Research Foundation, Lowell, MA
Lea Murphy, Math Dept., Carnegie-Hellon U., Pittsbury~), PA
James Nester, Physics Dept., U. of Maryland, College Park, MD
.A P~rez-Rendo~, Hath Dept., Univ. of Salamanca, Salamanca, Spain
Kathleen Pericak, Math Dept., Carnegie-Mellon U., Pittsburgh, PA
Tudor Rmtiu, MAth Dept, .U .C -Berkeley, Berkeley, CA
dL'waD Reynolds, #~ath Dept., Carne}<ie-Mellon U., Pittsburgh, PA
David Rod, Math Dept., .U of Calgary, Cal~7~ Canada
n]:wdE Lee Rogers, Math Dept., Rensselaer Polytechnic Institute, Troy, NY
P=[ch~d Sacksteder, CU~Y-Mathemat~es Graduate Center, W. 33 42 St., New York,
Willy Sarlet, Harvard U., Science Center, Cambridge, MA
Garfield Scb~Jdt, Math Dept., U. of Lowell, Lowell, ~A
Patrick Shanahan, Math Dept., College of the Holy Cross, Worcester, MA
Henry SLmpson, }~th Dept., U. of Tenm~essee, Y~oxville, TN
Marshall Slemrod, Math Dept., Rensselaer Polytechnic INstitute, Troy, NY
Scott Spector, Math Dept., U. of Tennessee, Knoxville, ~{
Gilbert Strang, Math Dept., M.I.T., Cambridge, MR
Leonard S~oAsk], Math Dept., College of the Holy Cross, Worcester, }%
E.R. Suryanarayan, Math Dept., .U of Rhode Island, Kingston, RI
Virginia Taylor, Math Dept., U. of Lowell, Lowell, MR
.W Tulczyjew, Math Dept., .U of Calgary, Canada
Philip Yasskin, Physics Dept., Harvard U., Cambridge, Y~
GEOMETRIC ASPECTS OF GLOBAL
BIFURCATION IN NONLINEAR ELASTICITY
by
Stuart .S Antman
Lefschetz Center for Dynamical Systems
Division of Applied Mathematics
Brown University
Providence, Rhode Island 02912
and
Department of Mathematics
University of Maryland
College Park, Maryland 20742
CONTENTS
i. Introduction ..................................... 1
2. The Equilibrium Equations for Nonlinearly
Elastic Rods ..................................... 3
3. The Buckling of a Straight Rod ................... 9
4. The Buckling of a Circular Plate ................. 16
5. The Buckling of a Circular Arch .................. 22
6. The Buckling of a Spherical Shell ................ 24
7. Other Problems ................................... 25
8. References ....................................... 28
.I Introduction
The science of continuum mechanics treats the change of shape of
material bodies under the action of forces. The basic problem of
continuum mechanics is to find the position ~(x,t) of each material
point x of a body B at each time t in some time interval, given the
nature of the material of B, the force intensity per unit volume of B,
and suitable conditions on the boundary ~B of B. (We may identify
the body B with the region of Euclidean 3-space ~3 it occupies in
some reference configuration and we may identify a material point with
its position in this configuration. The reference configuration could
be taken to be the configuration occupied by the body at some initial
time, or could be taken to be a natural state, which is one in which
the net force on each part of the body is zero.) Thus Euclidean
geometry, which is not as simple as it sounds, enters continuum
mechanics at the most primitive level. In this paper we examine some
of the ways this underlying Euclidean geometry interacts with the
geometrical or topological machinery used to analyze the governing
equations.
Before embarking on the main theme of this paper, which is the
study of this interaction in global bifurcation problems, it is worth-
while to pause to examine briefly a manifestation of this interaction
in another and more fundamental setting. The requirement that the
resultant force and moment on every (regular) subbody of B re-
spectively equal the time derivative of linear and angular momentum
yields the equations of motion for B. These involve the stress, which
is the intensity of force over surfaces. The material properties of B
are specified by constitutive equatigns, which prescribe the dependence
of the stress on the deformation ~p/~x. (~p/~x is the tensor whose
components are the partial derivatives of components of p with re-
spect to components of x.) The substitution of the constitutive
equations into the equations of motion yields the governing equations
for B. We represent these together with boundary, initial, and other
subsidiary conditions in the abstract form
{I.I) F(p) = .O
Here F is a quasilinear differential operator from one suitable
collection of functions to another. (We shall derive concrete realiza-
tions of (I.i) for elementary, but important, mechanical problems in
Section 2.) F may profitably be interpreted as an infinite-
dimensional vector field defined over its infinite-dimensional domain.
is a solution of (I.i) if the vector field F vanishes at p. In
a formal way we have thus introduced geometric notions into our
mechanical problem at a much deeper level.
Let us now examine how these two levels of geometric structure
interact. To ensure that B is not torn apart by the forces acting
on it, we require that p(.,t) be continuous for each .t To ensure
that two material points of B do not simultaneously occupy the same
point of space, we require that p(.,t) be invertible. But in-
vertibility is a global restriction p, which is also required to
satisfy local restrictions (i.i). Only now is work in analysis and
geometry beginning to confront questions like that of finding physically
reasonable restrictions on the data of (i.I) for its solutions to be
invertible. We are accordingly motivated to replace the global re-
quirement that p(.,t) be invertible by the local requirement that
~(.,t) preserve orientation, i.e., that itsJacobian be positive:
(1.2) det(~p/~x) > .0
There are some serious questions relating to the definition of this
Jacobian when p is not continuously differentiable, which we pass
over (cf. [16]). It is not difficult to show that the set of p's
satisfying (1.2), however it is defined, is neither convex nor closed
(cf. [2]). Thus (l.2),an innocent restriction arising from Euclidean
geometry, produces unpleasant geometric and topological consequences
for the domain of F. This suggests that analytic treatments of
problems involving (1.2) are likely to be difficult. This is indeed
the case. The analysis of one-dimensional static problems involving
(1.2), though delicate, is in a rather complete state (cf. [2,4,8])
but there are no corresponding results for one-dimensional dynamical
problems. Significant progress for three-dimensional static problems
has been made by Ball [16], yet much remains to be done on this very
deep problem. We remark that the serious analysis of (1.2) has only
recently occurred because only recently have scientists seriously
studied large deformations.
Analogs of (1.2) will appear in our work where they will play but
a subsidiary role. Our central concern is to describe recent work
on the application of topological methods to equations like (i.i) in
order to obtain detailed qualitative information about the deformed
shapes of elastic structures. We limit our attention to static problems
for nonlinearly elastic structures, especially bifurcation problems,
that are described by systems of ordinary differential equations. The
use of these equations not only enables us to avoid the severe technical
difficulties presented by related partial differential equations, but
also allows us to exploit the particularly rich analytic structure of
ordinary differential equations that underlies our qualitative analyses.
In Section 2 we derive the equations for the equilibrium of rods
because the derivation has two features that we wish to emphasize:
simplicity and the absence of ad hoc geometric approximations. In the
next four sections we discuss the buckling of straight rods under
terminal thrust, of circular plates under edge thrust, of circular
arches under hydrostatic pressure, and of spherical shells under hydro-
static pressure. We wish to examine the effects of the initial
curvature of the arch and shell and the effects of singularities due
to the use of polar coordinates for the plate and shell. Thereafter,
we discuss a number of other problems.
.2 The. Equilibrium Equations for Nonlinearly Elastic Rods
We adopt a mathematical model for the deformation of nonlinearly
elastic rods that is governed by a rich, yet tractable, quasilinear
system of ordinary differential equations. To motivate our model, we
