Table Of ContentTexts and
Monographs
in Physics
W. Beiglbock
E. H. Lieb
T. Regge
W. Thirring
Series Editors
Ruggero Maria Santi II i
Foundations of
Theoretical Mechanics II
Birkhoffian Generalization
of Hamiltonian Mechanics
[$]
Springer-Verlag
New York Heidelberg Berlin
Ruggero Maria Santilli
The Institute for Basic Research
96 Prescott Street
Cambridge, MA 02138
U.S.A.
Editors:
Wolf BeiglbOck Elliott H. Lieb
Institut fUr Angewandte Mathematik Department of Physics
Universitat Heidelberg Joseph Henry Laboratories
1m Neuenheimer Feld 5 Princeton University
D-6900 Heidelberg I P.O. Box 708
Federal Republic of Germany Princeton, NJ 08540
U.S.A.
Tullio Regge Walter Thirring
Universita di Torino Institut fUr Theoretische Physik
Istituto di Fisica Teorica der Universitat Wien
C.so M. d'Azeglio, 46 Boltzmanngasse 5
10125 Torino A-I090 Wien
Italy Austria
Library of Congress Cataloging in Publication Data
Santilli, Ruggero Maria, 1935-
Birkhoffian generalization of Hamiltonian mechanics.
(F oundations of theoretical mechanics; 2) (Texts
and monographs in physics)
Bibliography: p.
Includes index.
1. Mechanics. 2. Inverse problems (Differential equations) 3. Hamiltonian sys
tems. I. Title. II. Series: Santilli, Ruggero Maria, 1935- . Foundation oftheoreti
cal mechanics; 2. III. Series: Texts and monographs in physics.
QA805.S254 1978 vol. 2 531s [531J 82-19319
[QA808J
All rights reserved.
No part of this book may be translated or reproduced in any form without written
permission from Springer-Verlag, 175 Fifth Avenue, New York, New York 10010,
U.S.A.
© 1983 by Springer-Verlag New York Inc.
Softcover reprint of the hardcover 1s t edition 1983
Typeset by Composition House Limited, Salisbury, England.
9 8 7 6 5 432 I
ISBN-13: 978-3-642-86762-0 e-ISBN-13: 978-3-642-86760-6
001: 10.1007/978-3-642-86760-6
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Contents
Preface xi
Acknowledgments xix
Introduction 1
4 Birkhoff's Equations 12
4.1 Statement of the problem 12
4.2 Birkhoff's equations 30
4.3 Birkhoffian representations of Newtonian systems 42
4.4 Isotopic and genotopic transformations of first-order systems 46
4.5 Direct universality of Birkhoff's equations 54
Charts:
4.1 Lack of algebraic character of nonautonomous Birkhoff's equations
68
4.2 Algebraic significance of isotopic and genotopic transformations 70
4.3 Havas's theorem of universality of the inverse problem for systems of
arbitrary order and dimensionality 75
4.4 Rudiments of differential geometry 77
4.5 Global treatment of Hamilton's equations 81
4.6 Global treatment of Birkhoff's equations 85
4.7 Lie-admissible/symplectic-admissible generalization of Birkhoff's equa
tions for nonlocal non potential systems 90
Examples 98
Problems 108
vii
V1ll Contents
5 Transformation Theory of Birkhoff's Equations 110
5.1 Statement of the problem 110
5.2 Transformation theory of Hamilton's equations 114
5.3 Transformation theory of Birkhoff's equations 126
Charts:
5.1 Need to generalize the contemporary formulation of Lie's theory 148
5.2 Isotopic generalization of the universal enveloping associative algebra
154
5.3 Isotopic generalization of Lie's first, second, and third theorems 163
5.4 Isotopic generalizations of enveloping algebras, Lie algebras, and Lie
groups in classical and quantum mechanics 173
5.5 Darboux's theorem of the symplectic and contact geometries 184
5.6 Some definition of canonical transformations 187
5.7 Isotopic and genotopic transformations of variational principles 188
Examples 194
Problems 197
6 Generalization of Galilei's Relativity 199
6.1 Generalization of Hamilton-lacobi theory 199
6.2 Indirect universality of Hamilton's equations 217
6.3 Generalization of Galilei's relativity 225
Charts:
6.1 Applications to hadron physics 253
6.2 Applications to statistical mechanics 261
6.3 Applications to space mechanics 264
6.4 Applications to engineering 264
6.5 Applications to biophysics 267
Examples 270
Problems 277
Appendix A: Indirect Lagrangian Representations 281
A.I Indirect Lagrangian representations within fixed local variables 281
A.2 Isotopic transformations of a Lagrangian 289
A.3 Indirect Lagrangian representations via the use of the transformation theory
297
Charts:
A.I Analytic Newtonian systems 307
A.2 Analytic extensions of Lagrangian and Hamiltonian functions to
complex variables 309
A.3 The Cauchy-Kovalevski theorem 312
A.4 Kobussen's treatment of Darboux's theorem of universality for one-
dimensional systems 314
A.5 Vanderbauwhede's functional approach to the inverse problem 317
A.6 Symmetries 321
A.7 Lie's construction of symmetries of given equations of motion 324
A.8 First integrals and conservation laws 327
Contents ix
A.9 Noether's construction of first integrals from given symmetries 333
A.1O Isotopic transformations, symmetries, and first integrals 338
A.II Lack of a unique relationship between space-time symmetries and
physical laws 340
A.12 Classification of the breakings of space-time symmetries in Newtonian
mechanics 344
Examples 348
Problems 356
References 359
Index 365
Preface
In the preceding volume,l I identified necessary and sufficient conditions for
the existence of a representation of given Newtonian systems via a variational
principle, the so-called conditions of variational self-adjointness.
A primary objective of this volume is to establish that all Newtonian
systems satisfying certain locality, regularity, and smoothness conditions,
whether conservative or nonconservative, can be treated via conventional
variational principles, Lie algebra techniques, and symplectic geometrical
formulations. This volume therefore resolves a controversy on the repre
sentational capabilities of conventional variational principles that has been
lingering in the literature for over a century, as reported in Chart 1.3.1.2
The primary results of this volume are the following. In Chapter 4,3 I prove
a Theorem of Direct Universality of the Inverse Problem. It establishes the
existence, via a variational principle, of a representation for all Newtonian
systems of the class admitted (universality) in the coordinates and time
variables of the experimenter (direct universality). The underlying analytic
equations turn out to be a generalization of conventional Hamilton equations
(those without external terms) which: (a) admit the most general possible
action functional for first-order systems; (b) possess a Lie algebra structure
in the most general possible, regular realization of the product; and (c)
1 Santilli (1978a). As was the case for Volume I, the references are listed at the end of this
volume, first in chronological order and then in alphabetic order.
2 All references to the preceding volume have the prefix" I ", e.g., Section 1.1.1, Equation
(1.1.1.5). Script letter f is used to refer to elements within the Introduction to the present
volume.
3 To stress the continuity with the three chapters of Volume I, those of this volume are
numbered 4, 5, and 6.
Xl
xii Preface
characterize a symplectic two-form in its most general possible local and
exact formulation. For certain historical reasons, indicated in the text, I have
called these equations Birkhoff's equations.
In Chapter 5 I present the transformation theory of Birkhoff's equations.
Essentially, it emerges that, while Hamilton's equations preserve their
structure only under special classes of transformations (the canonical and
the canonoid), Birkhoff's equations preserve their structure under arbitrary,
generally non canonical, transformations. I then present a step-by-step
generalization of the Hamiltonian transformation theory. In addition, I
point out that Birkhoff's equations can be obtained from Hamilton's
equations via the use of non canonical transformations. The inverse reduction
occurs instead via the use of Darboux's transformations of the symplectic
geometry. This allows the proof in Chapter 6 of the Theorem of Indirect
Universality of Hamilton's Equations, according to which conventional
Hamilton equations are unable to represent Newtonian systems at large in
the reference frame of their experimental observation; nevertheless, a
representation can always be achieved via use of the transformation theory.
As has been known since Galilei's time, physics requires that abstract
mathematical algorithms admit a realization in the frame of the observer.
The inability of Hamilton's equations to satisfy this fundamental requirement
confirms the need for their Birkhoffian generalization.
The analysis presented in these volumes therefore establishes that the
treatment in the frame of the observer of Newtonian systems with unre
stricted dynamical conditions requires the use of generalized analytic
formulations for the most general possible first-order Pfaffian action and of
generalized geometric formulations for the most general possible local and
exact two-forms. These occurrences render inevitable a reinspection of Lie's
theory (enveloping associative algebras, Lie algebras, and Lie groups) to
achieve a form which is directly compatible with the generalized analytic
and geometric formulations-that is, a form which is classically of non
canonical character and quantum mechanically of predictable non unitary
character. This study is conducted in the final stage of a program where the
existence of generalized algebraic formulations is shown. These formulations
essentially consist of a reformulation of Lie's theory that is directly applicable
to the most general possible associative envelopes, the most general possible
non-Hamiltonian/Birkhoffian realizations of the Lie product, and the most
general possible noncanonical/nonunitary structures of the Lie groups. By
keeping in mind that Lie's theory was developed for the simplest possible
associative product XiXj of the envelope, the simplest possible form
Xi Xj - Xj Xi of the Lie product, and the simplest possible structure exp (}iXi
of the Lie groups, the need for the reformulation under consideration is self
evident. I have called the emerging formulations isotopic generalizations,
where the term" isotopic" expresses the preservation of the primary analytic,
Lie, or symplectic character.
In this way, we see the emergence of the foundations of a Birkhoffian
Generalization of Hamiltonian Mechanics which
Preface xiii
1. applies to a class of physical systems broader than that for which
Hamiltonian Mechanics was conceived-systems with action-at-a
distance, potential, self-adjoint forces, as well as contact, non
potential, non-self-adjoint forces;
2. is based on an isotopic generalization of the analytic, algebraic, and
geometric methods of Hamiltonian Mechanics; and
3. is capable of recovering Hamiltonian Mechanics identically when all
non-self-adjoint forces are null.
A number of applications to systems of ordinary differential equations in
Newtonian Mechanics, Space Mechanics, Statistical Mechanics, Engineering,
and Biophysics are presented during the course of our analysis, with more
specific treatment appearing in Chapter 6. With the understanding that
quantum mechanical profiles are beyond the scope of this volume, I have
briefly indicated the existence of an isotopic generalization of Heisenberg's
equations, as well as of a number of related quantum mechanical aspects, for
the description of particles under action-at-a-distance, potential interactions,
as well as contact, nonpotential interactions, which are conceivable under
mutual wave penetration, and overlap. The rather old (and currently dormant)
problem of the generalization of Quantum Mechanics is therefore brought
to life in an intriguing and direct way by the Birkhoffian Generalization of
Hamiltonian Mechanics. Regrettably, for the sake of brevity I have been
forced to ignore several additional, equally intriguing developments such as
the extension of Birkhoffian Mechanics to field theory-a study which has
already been initiated in the literature.4
The mathematically inclined reader should be informed from the outset
that I have given priority of presentation to methods and insights, not only
in local coordinates but also within a single fixed system of variables, those
relative to the observer. The use of transformation theory is presented only
as a second phase of study. Finally, generalization via coordinate-free, global,
and geometric approaches is presented as a more advanced approach. This
style of presentation implies a reversal of the priorities of contemporary
mathematical studies, particularly those of geometric character, but it is
dictated by specific pedagogical and technical needs.
On pedagogical grounds, my teaching experience has suggested that it is
best to expose students first to geometric structures in specific local variables
and show that the essential geometric properties persist under arbitrary (but
smoothness- and regularity-preserving) transformations of the local vari
ables. Then the students may be brought, in a progressive motivated way,
to advanced coordinate-free techniques.
The technical reasons for giving priority to formulating the methods in
local variables are even more pressing than the pedagogical ones. In fact,
the crucial inability of conventional Hamilton equations to represent New
tonian systems in the frame of the observer can be identified only via the
local formulation of the theory because, at the abstract, coordinate-free level,
4 Kobussen (1979).
