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A generalization of the Liouville–Arnol'd theorem

Published online by Cambridge University Press:  24 October 2008

G. E. Prince ,
G. B. Byrnes ,
J. Sherring  and
S. E. Godfrey
G. E. Prince
Affiliation:
Mathematics Department, La Trobe University, Bundoora, Victoria 3083, Australia
G. B. Byrnes
Affiliation:
Mathematics Department, La Trobe University, Bundoora, Victoria 3083, Australia
J. Sherring
Affiliation:
Mathematics Department, La Trobe University, Bundoora, Victoria 3083, Australia
S. E. Godfrey
Affiliation:
Mathematics Department, La Trobe University, Bundoora, Victoria 3083, Australia
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Abstract

We show that the Liouville-Arnol'd theorem concerning knowledge of involutory first integrals for Hamiltonian systems is available for any system of second order ordinary differential equations. In establishing this result we also provide a new proof of the standard theorem in the setting of non-autonomous, regular Lagrangian mechanics on the evolution space ℝ × TM of a manifold M. Both the original theorem and its generalization rely on a certain bijection between symmetries of the system and its first integrals. We give two examples of the use of the theorem for systems on ℝ2 which are not Euler-Lagrange.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 117 , Issue 2 , March 1995 , pp. 353 - 370
Copyright
Copyright © Cambridge Philosophical Society 1995

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