R. Paul
A KAM Theorem for Some Degenerate Hamiltonian Systems
(85K, Latex 2e)

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.