Foundations of Theoretical Mechanics I: The Inverse Problem in Newtonian Mechanics

282 Pages·1978·7.955 MB·English

by  Ruggero Maria Santilli (auth.)

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Ruggero Maria Santilli
Foundations of
Theoretical Mechanics I
The Inverse Problem in
Newtonian Mechanics
[I]
Springer Science+Business Media, LLC
Ruggero Maria Santilli
Lyman Laboratory of Physics
Harvard University
Cambridge, Massachusetts 02138
USA
Editors:
Wolf Beiglbock Maurice Goldhaber
Institut fUr Angewandte Mathematik Department of Physics
Universitiit Heidelberg Brookhaven National Laboratory
1m Neuenheimer Feld 5 Associated Universities, Inc.
Upton, NY 11973
D-6900 Heidelberg 1
USA
Federal Republic of Germany
Elliott H. Lieb Walter Thirring
Department of Physics Institut fUr Theoretische Physik
Joseph Henry Laboratories der Universitat Wien
Princeton University Boltzmanngasse 5
A-1090 Wien
P.O. Box 708
Princeton, NJ 08540
USA
ISBN 978-3-662-23683-3 ISBN 978-3-662-25771-5 (eBook)
DOI 10.1007/978-3-662-25771-5
Library of Congress Cataloging in Publication Data
Santilli, Ruggero Maria.
Foundations of theoretical mechanics.
(Texts and monographs in physics)
Bibliography: p.
Includes index.
I. Mechanics. 2. Inverse problems
(Differential equations) I. Title.
QA808.S26 531 78-9735
All rights reserved.
No part of this book may be translated or reproduced in
any form without written permission from Springer-Verlag.
Copyright © 1978 by Springer Science+Business Media New York
Originally published by Springer-Verlag New York, Inc. in 1978
Softcover reprint of the hardcover 2nd edition 1978
9 8 7 6 5 4 3 2 (Corrected second printing, 1984)
ilu.eoto votum.e .e d.ed~:Cato a m{a n'lOfllt".e
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Contents
Preface xi
Acknowledgments xv
Volume Organization xvii
Use Suggestions xix
Introduction 1
1 Elemental Mathematics 15
1.1 Existence theory for implicit functions, solutions, and derivatives in the
parameters 15
1.2 Calculus of differential forms, Poincare lemma, and its converse 26
1.3 Calculus of variations, action functional, and admissible variations 33
Charts:
1.1 A theorem on the existence, uniqueness, and continuity of the implicit
functions for Newtonian systems 44
1.2 A theorem on the existence, uniqueness, and continuity of a solution
of a Newtonian initial value problem 45
1.3 A theorem on the existence, uniqueness, and continuity ofthe derivatives
with respect to parameters of solutions of Newtonian systems 46
1.4 A relationship between local and global solutions for conservative
systems 47
1.5 Hilbert space approach to Newtonian Mechanics 47
Examples 49
Problems 52
2 Variational Approach to Self-Adjointness 54
2.1 Equations of motion, admissible paths, variational forms, adjoint systems
and conditions of self-adjointness 54
VII
viii Contents
2.2 Conditions of self-adjointness for fundamental and kinematical forms of
Newtonian systems 63
2.3 Reformulation of the conditions of self-adjointness within the context of
the calculus of differential forms 70
2.4 The problem of phase space formulations 76
2.5 General and normal forms of the equations of motion 78
2.6 Variational forms of general and normal systems 82
2.7 Conditions of self-adjointness for general and normal systems 85
2.8 Connection with self-adjointness of linear operators 90
2.9 Algebraic significance of the conditions of self-adjointness 92
Charts:
2.1 Hausdorff, second-countable, oo-differentiable manifolds 96
2.2 Newtonian systems as vector fields on manifolds 97
2.3 Symplectic manifolds 99
2.4 Contact manifolds 10 I
2.5 Geometrical significance of the conditions of self-adjointness 102
Examples 103
Problems 107
3 The Fundamental Analytic Theorems
of the Inverse Problem 110
3.1 Statement of the problem 110
3.~ The conventional Lagrange's equations 112
3.3 Self-adjointness of the conventional Lagrange's equations 116
3.4 The concept of analytic representation in configuration space 119
3.5 The fundamental analytic theorem for configuration space formula
tions 130
3.6 A method for the construction of a Lagrangian from the equations of
motion 141
3.7 The implications of nonconservative forces for the structure of a
Lagrangian 147
3.8 Direct and inverse Legendre transforms for conventional analytic rep
resentations 157
3.9 The conventional Hamilton's equations 166
3.10 Self-adjointness of the conventional Hamilton's equations 169
3.11 The concept of analytic representation in phase space 173
3.12 The fundamental analytic theorem for phase space formulations and a
method for the independent construction of a Hamiltonian 176
Charts:
3.1 The controversy on the representation of nonc onservative Newtonian
systems with the conventional Hamilton's principle 182
3.2 The arena of applicability of Hamilton's principle 183
3.3 Generalization of Hamilton's principle to include the integrability
conditions for the existence of a Lagrangian 185
3.4 Generalization of Hamilton's principle to include Lagrange's equa
tions and their equations of variation 187
3.5 Generalization of Hamilton's principle to include Lagrange's equa
tions, their equations of variations, and the end points con
tributions 188
3.6 Generalization of Hamilton's principle to include a symplectic
structure 189
Contents ix
3.7 Generalization of Hamilton's principle for the unified treatment of
the Inverse Problem in configuration and phase space 191
3.8 Self-adjointness of first-order Lagrange's equations 192
3.9 The fundamental analytic theorem for first-order equations of
motion in configuration space 194
3.10 A unified treatment of the conditions of se\f-adjointness for first-,
second-, and higher-order ordinary differential equations 195
3.11 Engels' methods for the construction of a Lagrangian 196
3.12 Mertens' approach to complex Lagrangians 198
3.13 Bateman's approach to the Inverse Problem 199
3.14 Douglas' approach to the Inverse Problem 200
3.15 Rapoport's approach to the Inverse Problem 201
3.16 Vainberg's approach to the Inverse Problem 203
3.17 Tonti's approach to the Inverse Problem 204
3.18 Analytic, algebraic and geometrical significance of the conditions of
variational self-adjointness 205
Examples 206
Problems 215
Appendix: Newtonian Systems 219
A.1 Newton's equations of motion 219
A.2 Constraints 220
A.3 Generalized coordinates 223
AA Conservative systems 224
A.5 Dissipative systems 228
A.6 Dynamical systems 233
A.7 The fundamental form of the equations of motion in configuration
space 235
Charts:
AI Galilean relativity 237
A2 Ignorable coordinates and conservation laws 239
A3 Impulsive motion 240
AA Arrow of time and entropy 241
A.5 Gauss principle of least constraint 242
A6 The Gibbs-Appel equations 243
A7 Vi rial theorem 243
A8 Liouville's theorem for conservative systems 244
A9 Generalizations of Liouville's theorem to dynamical systems 244
AlO The method of Lagrange undetermined multipliers 246
A.ll Geometric approach to Newtonian systems 247
Al2 Tensor calculus for linear coordinate transformations 248
A.13 Tensor calculus for nonlinear coordinate transformations 249
A.14 Dynamical systems in curvilinear coordinates 250
Examples 252
Problems 254
References 257
Index 262
Preface
The objective of this monograph is to present some methodological
foundations of theoretical mechanics that are recommendable to graduate
students prior to, or jointly with, the study of more advanced topics such
as statistical mechanics, thermodynamics, and elementary particle physics.
A program of this nature is inevitably centered on the methodological
foundations for Newtonian systems, with particular reference to the central
equations of our theories, that is, Lagrange's and Hamilton's equations.
This program, realized through a study of the analytic representations in
terms of Lagrange's and Hamilton's equations of generally nonconservative
Newtonian systems (namely, systems with Newtonian forces not necessarily
derivable from a potential function), falls within the context of the so-called
Inverse Problem, and consists of three major aspects:
l. The study of the necessary and sufficient conditions for the existence
of a Lagrangian or Hamiltonian representation of given equations of
motion with arbitrary forces;
2. The identification of the methods for the construction of a Lagrangian
or Hamiltonian from given equations of motion verifying conditions
1; and
3 The analysis of the significance of the underlying methodology for
other aspects of Newtonian Mechanics, e.g., transformation theory,
symmetries, and first integrals for nonconservative Newtonian
systems.
This first volume is devoted to the foundations of the Inverse Problem,
with particular reference to aspects I and 2. The second volume deals
with some generalizations and applications of the Inverse Problem, with
Xl