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On the number of caustics for invariant tori of Hamiltonian systems with two degrees of freedom

Published online by Cambridge University Press:  19 September 2008

Misha Bialy
Misha Bialy
Affiliation:
Department of Theoretical Mathematics, The Weizmann Institute of Science, Rehovot 76100, Israel
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Extract

Let X be a two-dimensional orientable connected manifold without boundary, H: T*X → ℝ a smooth hamiltonian function denned on the cotangent bundle. We will assume that H is of a ‘classical type’ that is convex and even on each fibre Tx*X. The goal of this paper is to describe the set Γ of all singular points of the projection Θ|L where ι: L → T*X is a smooth embedded 2-torus invariant under the hamiltonian flow h1, Θ: T*X → X is the canonical projection.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 11 , Issue 2 , June 1991 , pp. 273 - 278
Copyright
Copyright © Cambridge University Press 1991

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References

REFERENCES

[1]Cushman, R. & Duistermaat, J.J.. The quantum mechanical spherical pendulum. Bull. Amer. Math. Soc. 19 (2) (1988), 475–479.Google Scholar
[2]Arnold, V.I.. Mathematical Methods of Classical Mechanics. Springer: Berlin, Heidelberg, New York (1978).Google Scholar
[3]Herman, M.. Sur les courbes invariantes par les difféomorphisms de l'anneau. Astérisque 103–104 (1983).Google Scholar
[4]Bialy, M.L. & Polterovich, L.V.. Lagrangian singularities of invariant tori of hamiltonian systems with two degrees of freedom. Invent. Math. 97 (1989), 291–303.Google Scholar
[5]Bialy, M.L.. Aubry—Mather sets and Birkhoff's theorem for geodesic flows on the two-dimensional torus. Comm. Math. Phys. 126 (1989), 13–24.Google Scholar
[6]Moser, J.. Monotone twist mappings and the calculus of variations. Ergod. Th. & Dynam. Sys. 6 (1986), 401–413.CrossRefGoogle Scholar
[7]Toda, M.. Theory of Nonlinear Lattices. Springer: Berlin, Heidelberg, New York (1981).CrossRefGoogle Scholar