Methods for the primitive and uniform semiclassical quantization
of Nonseparable Hamiltonian Systems are developed. Integrable
nonseparable Hamiltonians are transformed to angle-action variables
by a sequence of Van Vleck t~ansformations. This sequence of transformations
displays accelerated convergence. The primitive semiclassical
eigenvalues are found by evaluating the transformed
Hamiltonian at the quantized values of the actions. For the nonintegrable
problem, an integrable approximation is found by normalizing
the Hamiltonian. This is accomplished by a sequence of
Birkoff-Van Vleck transformations which display accelerate convergence.
The uniform semiclassical eigenvalues are found by numerical
evaluation of the uniform semiclassical quantization condition.
These methods have been applied to the degenerate HenonHeiles
Hamiltonian with excellent results.