Foundations of Theoretical Mechanics I The Inverse Problem in Newton
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,
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W. BeiglbOck M. Goldhaber E. H. Lieb W. Thirring Series Editors
Ruggero Maria Santilli
Foundations of Theoretical Mechanics I The Inverse Problem in Newtonian Mechanics
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Springer-Verlag
New York
Heidelberg Berlin
Ruggero Maria Santilli Lyman Laboratory of Physics Harvard University Cambridge, Massachusetts 02138 USA
Editors:
Wolf Beiglböck
Maurice Goldhaber
Institut für Angewandte Mathematik Universität Heidelberg Im Neuenheimer Feld 5 D-6900 Heidelberg 1 Federal Republic of Germany
Department of Physics Brookhaven National Laboratory Associated Universities, Inc. Upton, NY 11973 USA
Elliott H. Lieb
Walter Thirring
Department of Physics Joseph Henry Laboratories Princeton University P.O. Box 708 Princeton, NJ 08540 USA
Institut für Theoretische Physik der Universität Wien Boltzmanngasse 5 A-l090.Wien
ISBN 978-3-642-86759-0 ISBN 978-3-642-86757-6 (eBook) DOI 10.1007/978-3-642-86757-6
Library of Congress Cataloging in Publication Data Santilli, Ruggero Maria. F oundations of theoretical mechanics. (Texts and monographs in physics) Bibliography: p. Includes index. 1. 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-Verlag New York Inc. Softcover reprint of the hardcover 1st edition 1978
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Contents
Preface Acknowledgments Volume Organization Use Suggestions Introduction
1
Elemental Mathematics
xi xv xvii xix 1
15
1.1 Existence theory for implicit functions, solutions, and derivatives in the parameters I 5 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-adjointnes
