PROCEEDINGS OF THE EIGHTH SYMPOSIUM IN APPLIED MATHEMATICS OF THE AMERICAN MATHEMATICAL SOCIETY. Held at the University of Chicago April 12-13, 1956

Size: px

Start display at page:

Download "PROCEEDINGS OF THE EIGHTH SYMPOSIUM IN APPLIED MATHEMATICS OF THE AMERICAN MATHEMATICAL SOCIETY. Held at the University of Chicago April 12-13, 1956"

Rudolph Riley
5 years ago
Views:

2 PROCEEDINGS OF THE EIGHTH SYMPOSIUM IN APPLIED MATHEMATICS OF THE AMERICAN MATHEMATICAL SOCIETY Held at the University of Chicago April 12-13, 1956 COSPONSORED BY THE OFFICE OF ORDNANCE RESEARCH Lawrence M. Graves EDITOR Prepared by the American Mathematical Society under Contract No. DA ORD-3777 with the Ordnance Corps, U.S. Army. Copyright 1958 by the McGraw-Hill Book Company, Inc. Printed in the United States of America. All rights reserved except those granted to the United States Government. Otherwise, this book, or parts thereof, may not be reproduced in any form without permission of the publishers. Library of Congress Catalog Card Number Reprinted 1986 The paper used in this journal is acid-free and falls within the guidelines established to ensure permanence and durability.

3 PROCEEDINGS OF SYMPOSIA IN APPLIED MATHEMATICS VOLUME VIII CALCULUS OF VARIATIONS AND ITS APPLICATIONS McGRAW-HILL BOOK COMPANY, INC. NEW YORK TORONTO LONDON 1958 FOR THE AMERICAN MATHEMATICAL SOCIETY

4 CONTENTS EDITOR'S PREFACE V On Variational Principles in Elasticity 1 BY ERIC REISSNER Variational Principles in the Mathematical Theory of Plasticity. BY D. C. DRUCKER Discussion of D. C. Drucker's Paper "Variational Principles in the Mathematical Theory of Plasticity" 23 BY P. G. HODGE, JR. A Geometrical Theory of Diffraction 27 BY JOSEPH B. KELLER Upper and Lower Bounds for Eigenvalues 53 BY J. B. DIAZ Stationary Principles for Forced Vibrations in Elasticity and Electromagnetism.. 79 BY J. L. SYNGE A Variational Computation Method for Forced-vibration Problems 89 BY H. F. WEINBERGER Applications of Variational Methods in the Theory of Conformal Mapping BY M. M. SCHIPPER Dynamic Programming and Its Application to Variational Problems in Mathematical Economics 115 BY RICHARD BELLMAN Variational Methods in Hydrodynamics 139 BY S. CHANDRASEKHAR Some Applications of Functional Analysis to the Calculus of Variations BY E. H. ROTHE INDEX 153 iii

5 EDITOR'S PREFACE This volume contains the papers presented at the Eighth Symposium in Applied Mathematics, sponsored by the American Mathematical Society and the Office of Ordnance Research, and devoted to The Calculus of Variations and Its Applications. In addition to the nine invited addresses, there are included two brief notes, by P. G. Hodge, Jr., and by H. F. Weinberger, which were invited by the Program Committee and which embody discussion of the papers by D. C. Drucker and by J. L. Synge, respectively. It seems obvious that one symposium could not profitably pay attention to all the directions in which variational methods have been applied. From the consultations of the Program Committee there resulted a group of addresses principally directed to applications in dynamics, but treating several other topics also. The editor wishes to make special acknowledgment to the McGraw-Hill Book Company for their care in the production of the volume, and to all the authors for the careful preparation of their manuscripts. As a result the editor's task has been a comparatively light one. LAWRENCE M. GRAVES Editor

6 INDEX Allocation processes, multistage, 121 Aronszajn, N., 54 Bottleneck process, 134 Caustic, 35 Characteristic-value problem, double, 139 Courant, 55 Decision processes, continuous, 118 multistage, 115 Deformation theories, 12 Eiconal equation, 27, 40 Eiconal function, complex, 43 Eigenvalues, 105, 150 Electromagnetic vibrations, 85 Energy, complementary, 9 potential, 9 Equilibrium-stress field, 8 Euler (differential) equations, 2 Existence proof, 143 Fermat's principle, 38, 41 Green's function, 97 Hadamard variational formula, 102, 109 Hilbert transform, 108 Index form, 147 Instability, 139 Interior variation, method of, 97 Malus, theorem of, 40 Mappings, gradient, 149 Metric, definite, 79 indefinite, 79 Operators, half-bounded, 143 Optimal policy, 117 Perfectly plastic material, 18 Rate principle, complementary, 15 Rayleigh-Ritz method, 57 Rays, diffracted, 27, 28 imaginary, 41 normal congruences of, 40 Smoothing processes, 130 SnelPs law, 29, 39 Spectral theorem, 143 Stationary principles, 79, 89 Strain-displacement field, 8 Univalent functions, group property of, 94 Variational principle, for displacements, 3 for stresses, 4 Vibrating membrane, 80 Vibrations of an elastic body, 83 Weinstein's method, 60 Work hardening, 15 Work-hardening relations, 13 DEFGHIJ-AMS-89876

Mathematics for Economics

Mathematics for Economics third edition Michael Hoy John Livernois Chris McKenna Ray Rees Thanasis Stengos The MIT Press Cambridge, Massachusetts London, England c 2011 Massachusetts Institute of Technology