Foundations of Theoretical Mechanics II Birkhoffian Generalizations
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
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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
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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 Universitat Heidelberg 1m Neuenheimer Feld 5 D-6900 Heidelberg I Federal Republic of Germany
Department of Physics Joseph Henry Laboratories Princeton University P.O. Box 708 Princeton, NJ 08540 U.S.A.
Tullio Regge
Walter Thirring
Universita di Torino Istituto di Fisica Teorica C.so M. d'Azeglio, 46 10125 Torino Italy
Institut fUr Theoretische Physik der Universitat Wien Boltzmanngasse 5 A-I090 Wien Austria
Library of Congress Cataloging in Publication Data Santilli, Ruggero Maria, 1935Birkhoffian 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 systems. I. Title. II. Series: Santilli, Ruggero Maria, 1935- . Foundation oftheoretical 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 1st 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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Representation via Birkhoff's equations (a = (r, p»
h(3)
h(Z»)( t" - p'/mk )} • = h(4) Pk - f. - Fk SA
Self-adjoint general form
1
Pk - [k(t, r, p) - Fk(t, r, p)
{(ll
(
1
t, t)]SA -
Non-self-adjoint normal form
{[mki'k - Mt,
Essentially non-self-adjoint Newtonian systems
=
at
oR.
n.o = -no.
R. da· - B dt
nz = dRio
Rl =
0, 1,
da>,
oRp oR. Q.P = oa· - oaP' ~
oa·
v
1\
2, ... , 6N,
ji,
da P
=-+-
oB
0,
= (t, r, p);
noo =
a
p>
Contact geometry
nz = !n
V>
::l
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~
~
..0
tTl
V>
o ~
::r
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co ::;.
0\ 0\
Figure4.1 (A schematic view of the direct universality of Birkhoff's equations for local Newtonian systems). The physical systems in our environment (such as motions in atmosphere, spinning tops with drag torques, damped oscillations, etc.) generally violate the integrability conditions for the existence of a Hamiltonian representation in the coordinate and time variables of the observer. This fact, established in Section 4.1, is clear evidence of the insufficiency in mechanics of Hamilton's equations in their contemporary formulation (that without external terms). A central objective of this volume is the identification of a generalization of Hamilton's equations which
