98-19 R. Paul

A KAM Theorem for Some Degenerate Hamiltonian Systems (85K, Latex 2e) Jan 15, 98

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Abstract. We prove the existence of a large measure of invariant tori for a class of perturbed integrable systems in which the unperturbed Hamiltonian is degenerate (that is, its Hessian matrix does not have full rank). This class consists of systems for which--along with certain other restrictions--the unperturbed Hamiltonian added to the average of the perturbation is nondegenerate. Our result is similar to a theorem of Arnold, whose proof is only sketched. In addition, we obtain explicit estimates on the measure of phase space not foliated by invariant tori. We apply our result to a system designed by Weinberg to test the current theory of quantum mechanics. The system is a perturbed simple harmonic oscillator, to which KAM theorems with nondegeneracy conditions only on the unperturbed Hamiltonian do not apply.

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