Texts and Monographs in Physics

w. Beiglbock M. Goldhaber

E. H. Lieb W. Thirring

Series Editors

Texts and Monographs in Physics

R. Bass: Nuclear Reactions with Heavy Ions (1980). A. Bohm: Quantum Mechanics (1979). O. Bratteli and D.W. Robinson: Operator Algebras and Quantum Statistical

Mechanics. Volume I: C· -and W· -Algebras. Symmetry Groups, Decomposition of States (1979). Volume II: Equilibrium States. Models in Quantum Statistical Mechanics (1981).

K.Chadan and P.C. Sabatier: Inverse Problems in Quantum Scattering Theory (1977).

G. Gallavotti: The Elements of Mechanics I (1983). W. GlOckle: The Quantum Mechanical Few-Body Problem (1983). J .M. Jauch and F. Rohrlich: The Theory of Photons and Electrons: The

Relativistic Quantum Field Theory of Charged Particles with Spin One-half, Second Expanded Edition (1980).

J. Kessler: Polarized Electrons (1976). G. Ludwig: Foundations of Quantum Mechanics 1(1983). R.G. Newton: Scattering Theory of Waves and Particles, Second Edition (1982). H. Pilkuhn: Relativistic Particle Physics (1979). R.D. Richtmyer: Principles of Advanced Mathematical PhYsics. Volume I (1978).

Volume II (1981). W. Rindler: Essential Relativity: Special, General, and Cosmological, Revised

Second Edition (1980). P.Ring and P. Schuck: The Nuclear Many-Body Problem (1980). R.M. Santilli: Foundations of Theoretical Mechanics. Volume I: The Inverse

Problem in Newtonian Mechanics (1978). Volume II: Birkhoffian Generalization of Hamiltonian Mechanics (1983).

M.D. Scadron: Advanced Quantum Theory and Its Applications Through Feynman Diagrams (1979).

C. Truesdell and S. Bharatha: The Concepts and Logic of Classical Thermodynamics as a Theory of Heat Engines: Rigourously Constructed upon the Foundation Laid by S. Carnot and F. Reech (1977).

F.J. Yndurain: Quantum Chromodynamics: An Introduction to the Theory of Quarks and Gluons (1983).

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 Institut fUr Angewandte Mathematik Universitiit Heidelberg 1m Neuenheimer Feld 5 D-6900 Heidelberg 1 Federal Republic of Germany

Elliott H. Lieb Department of Physics Joseph Henry Laboratories Princeton University P.O. Box 708 Princeton, NJ 08540 USA

Maurice Goldhaber Department of Physics Brookhaven National Laboratory Associated Universities, Inc. Upton, NY 11973 USA

Walter Thirring Institut fUr Theoretische Physik

der Universitat Wien Boltzmanngasse 5 A-1090 Wien

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)

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Contents

Preface Acknowledgments Volume Organization Use Suggestions

xi xv

xvii 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

3

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

The Fundamental Analytic Theorems of the Inverse Problem

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

110

3.5 The fundamental analytic theorem for configuration space formula­tions 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 rep­resentations 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 non conservative 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 con­tributions 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

Index

257

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

xii Preface

particular reference to aspect 3, and the problem of the construction of indirect analytic representation, i.e., of Lagrangians and Hamiltonians for systems which, in the given form, violate conditions 1 above.

I had several motivations for undertaking this task. The first motivation came to me as a teacher. Indeed, the decision to study the analytic representa­tions of systems with arbitrary Newtonian forces grew out of my uneasiness in teaching a graduate course in classical mechanics in the conventional manner. Typically, an articulated body of interrelated methodological formulations (i.e., analytic, variational, algebraic, geometrical, etc.) is presented; but in the final analysis, in view of the lack of knowledge of the methods for computing a Lagrangian for systems with more general Newtonian forces, these formulations are nowadays applicable only to systems with forces derivable from a potential function (basically, con­servative systems). My uneasiness was ultimately due to the fact that, strictly speaking, conservative systems do not exist in our Newtonian environment. As a result, the Lagrangian representation of conservative Newtonian systems is, in general, only a crude approximation of physical reality.

A few remarks are sufficient to illustrate this point. For instance, the entire conventional theory of the Lagrangian representation in the space of the generalized coordinates of conservative Newtonian systems subject to holonomic constraints, is based on the often tacit assumption that the constraints are frictionless. But in practice, holonomic constraints are realized by mechanical means, e.g., hinges, rods, etc. Therefore, the presence of frictional forces is inevitable whenever holonomic constraints occur and, in turn, a Lagrangian representation that does not reflect this dissipative nature can only be considered a first approximation of the systems considered

Owing to the fundamental nature of the knowledge of a Lagrangian or Hamiltonian, the above limitation of the conventional approach to Newtonian Mechanics is present at virtually all levels of the theory. For instance, the theory of canonical transformations for the one-dimensional harmonic oscillator is well known. But the extension of this theory to the more realistic case of the damped oscillator is not treated in currently available textbooks, again because ofthe lack of methods for constructing a Hamiltonian when damping forces are present. Similarly, the Hamilton­Jacobi theory of the frictionless spinning top is well known, but its extension to the system which actually occurs in our environment, namely, the spinning top with damping torque, is unknown at this time to the best of my knowl­edge. Therefore, the analysis presented in this monograph, the analytic representations of nonconservative Newtonian systems, grew out of my attempts to more closely represent Newtonian reality.

Other motivations for undertaking this task came to me as a theoretical physicist. As we all know, the significance of Newtonian Mechanics goes beyond the pragmatic aspect of merely studying Newtonian systems, because its methodological foundations apply, apart from technical rather than conceptual modifications, to several branches of physics, such as quantum mechanics and elementary particle physics. As soon as I became aware of

Preface Xlll

new methodological prospectives within the context of purely Newtonian systems, I became intrigued by their possible significance for other branches of physics.

Predictably, it will take a considerable amount of time and effort by more than one researcher to ascertain the possible significance of the Inverse Problem for non-Newtonian frameworks. Nevertheless, to stimulate research along these lines, a few remarks are presented in the Introduction.

Owing to the lack of recent accounts of the Inverse Problem in both the mathematical and the physical literature, one of the most time-consuming parts of my program has been the identification of the prior state of the art. Indeed, it was only after a laborious library search, which I conducted over a three-year period by moving backward in time to the beginning of the past century, that I came to realize that the methodological foundations of the Inverse Problem were fully established in the mathematical literature by the first part of this century within the context of the calculus of variations. This was the result of the contributions of several authors, such as Jacobi (1837), Helmholtz (1887), Darboux (1891), Mayer (1896), Hirsh (1898), Bohem (1900), K6nisberger (1901), Hamel (1903), Kurshak (1906), and others. The most comprehensive account of which I am aware is the thesis of D. R. Davis in 1926 at the Department of Mathematics of the University of Chicago, under the supervision of G. A. Bliss, subsequently expanded and published in three articles in 1928, 1929, and 1931 (see References). Since that time, regrettably, the problem remained largely ignored in both the mathematical and physical literature, with only a few exceptions known to me, which I shall indicate in the Introduction.

In this volume I present the results of my search in specialized mathemati­cal and physical literature, and of my efforts on aspects such as the use of the Converse of the Poincare Lemma for the proof of the central theorem on the necessary and sufficient conditions for the existence of a Lagrangian, the methods for the construction of a Lagrangian from the given equations of motion, the independent treatment of the Inverse Problem for phase space formulations without the prior knowledge of a Lagrangian, and the alge­braic or geometrical significance of the necessary and sufficient conditions for the existence of a Hamiltonian. However, owing to the vast accumulation of literature in classical mechanics, calculus of variations, and other disciplines over the centuries, I make no claim to originality.

I make no claim to mathematical rigor, either. I concentrated my efforts primarily on presenting and illustrating the basic concepts in as simple a manner as possible. In essence, by specific intent, this volume should be readable by first- or second-year graduate students without major diffi­culties. In writing this monograph, I have also attempted to render it self­sufficient-extensive reference study is needed only for certain complemen­tary aspects, such as for certain problems of the theory of differential equations or for certain geometrical interpretations, and a sound knowledge of undergraduate mechanics is the only prerequisite.

I have also made an effort to adopt the most widely used notations and symbols. When necessary, new notations are identified by footnotes.

xiv Preface

Equations are referred to by notation of the form (1.2. 3a), where 1, 2, and 3a indicate the chapter, the section, and the equations therein, respectively.

The references are listed at the end of the volume in chronological and then alphabetical order. My list of textbooks must be considered as purely representative, though incompletely so, of contributions in theoretical mechanics and related disciplines. However, for specialized topics not treated in currently available textbooks, I have listed all the relevant ref­erences of which I am aware.

January 3, 1978 Lyman Laboratory of Physics Harvard University Cambridge, Massachusetts

Note to the Corrected Second Printing

Ruggero Maria Santilli

The only modifications implemented in this edition relate to minor errors and/or misprints in formulae, dates, and language of the first printing. All concepts, methods, and theorems needing more elaboration have been reconsidered in Volume II, either in the main text or in footnotes.

These guidelines apply in particular to the construction of analytic representations of non-self-adjoint Newtonian systems, which is a main objective of Volume II and not of this volume. The same guidelines, therefore, hold for a variety of related notions, such as that of ordered indirect analytic representations.

I am deeply grateful to several colleagues for the courtesy of pointing out to me errors and/or misprints in the first printing, particularly to Professor L. Y. Bahar of Drexel University, Philadelphia, Pennsylvania, for a very careful reading of the volume. It is a pleasure also to express my sincere appreciation to Professor R. Trostel of the Technische UniversiUit Berlin, West Germany, and to Professor A. Schoeber of the Institute for Basic Research, Cambridge, Massachusetts, for equally invaluable suggestions.

February 9, 1984

Cambridge, Massachusetts Ruggero Maria Santilli

Acknowledgments

I simply have no words to express my gratitude to A. Shimony. It is a truism to say that, without his encouragement, support, and advice, this work would not have been completed.

I would also like to express my sincere gratitude to A. C. Hurst, S. Shanmugadhasan, and P. L. Huddleston for carefully reading an earlier version of the manuscript and for numerous suggestions.

I am also sincerely grateful to my graduate students J. Eldridge and A. Sen for carefully studying an earlier version. Their various suggestions were very valuable in the ,)Verall improvement of the manuscript.

Correspondence on several histori~al or technical points with H. Corben, P. Dedecker, D. G. B. Edelen, G. M. Ewing, A. Harvey, P. Havas, R. Herman, M. R. Hestenes, C. B. Morrey, L. A. Pars, E. G. Saletan, D. C. Spencer, P. Stehle and K. R. Symon must be acknowledged.

Special thanks must be acknowledged to:

R. Mertens, E. Engels, F. Cantrijn, and W. Sarlet. Their frequent assistance has simply been invaluable for the entire project. H. Rund and D. Lovelock. Their advice on the use of the calculus of differential forms has been invaluable for the proof of the main theorems. A. Thellung and J. Kobussen. Their suggestions and comments have also been invaluable. L. Y. Bahar and R. Brooks. Their critical reading of the manuscript has been invaluable. C. N. Ktorides. His penetrating critical comments have been equally invaluable.

It is a pleasure to acknowledge support from the U.S. Department of Energy under contract no. ER-78-S-02-47420.AOOO.

xv

xvi Acknowledgments

It is a pleasure to thank F. E. Low, H. Feshbach, and R. Jackiw for their hospitality at the Center for Theoretical Physics of the Massachusetts Institute of Technology in 1976-1977, where part of this project was con­ducted.

It is a pleasure to thank S. Weinberg, M. Tinkham, and H. Georgi for their hospitality at the Lyman Laboratory of Physics, where this project was completed. In particular, I would like to express my appreciation for the opportunity of delivering an informal seminar course on the Inverse Problem at the Lyman Laboratory during the fall of 1977, which proved to be invaluable for the finalization of this project.

D. Nordstrom gave generously of his time and experience in assisting with the editorial preparation of the manuscript.

I would like also to express my gratitude to the Editorial Staff of Springer­Verlag for their invaluable assistance in the finalization of this project.

Almost needless to say, I am solely responsible for the content of this volume, including several modifications implemented in the final version of the manuscript.

Volume Organization

In the Introduction, I formulate the Inverse Problem of the calculus of variations, point out its reduction to a Newtonian context, and indicate all the relevant references on such problem of which I am aware.

In Chapter 1, I outline the rudiments of three disciplines, ordinary differential equations, calculus of differential forms, and calculus of vari­ations, which are prerequisites for the methodology of the Inverse Problem.

In Chapter 2, I introduce the central mathematical tool of the analysis, the so-called variational approach to self-adjointness, and I specialize it to the most important forms of Newtonian systems.

In Chapter 3, I work out the central objectives of this monograph, which consist of the necessary and sufficient conditions for the existence of a Lagrangian or, independently, a Hamiltonian, the methods for their com­putation from given equations of motion, and an analysis of those Newtonian forces that are admissible by a Lagrangian or Hamiltonian representation.

In Appendix A, I review those concepts of Newtonian Mechanics that are useful for the analysis of the main text, to avoid excessive reference to the existing literature.

The presentation is organized into a main text, a series of charts, a set of examples, and problems. In the main text, I treat the essential concepts and formulations of the approach. In the charts, I present those complementary aspects which, even though not essential for the basic lines of the approach, are valuable for a deeper insight, and I touch on topics of more advanced nature for subsequent study by the interested reader. The examples are intended to illustrate the basic concepts introduced in the text only. The

xvii

xviii Volume Organization

problems are intended to test the student's understanding of the given methodology and then his capability to work out specific applications.

The generalization of methodology of the Inverse Problem for the con­struction of a Lagrangian or Hamiltonian representation of systems of ordinary differential equations which, as given, violate the integrability conditions, is treated in Santilli (1979).

Use Suggestions

This book can be used as a textbook for a one-term graduate course on the Inverse Problem or on Nonconservative Newtonian Mechanics.

For the use of this book as a reference book for a section of a regular graduate course in classical mechanics devoted to the Inverse Problem, the instructor is recommended to work out a summary presentation of Chapters 2 and 3.

For graduate students in physics, I recommend first reading the main text, verifying the illustrative examples, and working out the problems. Subsequent study should then incorporate the charts and quoted references. A prior inspection of the Appendix A might be recommendable.

For graduate students and instructors in mathematics, this book can be complemented by currently available treatises on the calculus of variations, optimal control theory, differential geometry, and other topics to formulate the Inverse Problem in these disciplines.

xix