| We use the technique of calculating progressively higher order normal forms to solve problems in two classes of Hamiltonian systems. Both of these systems have the property that the natural map between the "momenta" in the system and the "frequencies" of a nearby integrable system is not a diffeomorphism. Such systems are called "degenerate".;Our first result is an extension of the theorem of Kolmogorov, Arnol'd, and Moser. We prove the existence of a large measure of invariant tori for a class of perturbed integrable systems in which the unperturbed Hamiltonian has a Hessian matrix of less than 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 non-degenerate.;Our result is similar to a theorem of Arnol'd, 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 non-degeneracy conditions only on the unperturbed Hamiltonian do not apply.;Our second result concerns sufficient conditions for the integrability of a Hamiltonian flow in a neighborhood of a fixed point. We use an idea about "obstructions to integrability", originally formulated by Poincare and recently developed by de la Llave. We show that if all countably many such obstructions vanish in a neighborhood of a fixed point, and a "non-degeneracy condition" is satisfied away from the fixed point, then there is a smooth integrating transformation.;We then show that in certain restricted classes of systems the presence of as many integrals in involution as degrees of freedom implies that all the obstructions vanish. |
