Remark on integrable Hamiltonian systems

An extension ton degrees of freedom of the fact is established that forn=1 the time and the energy constant are canonically conjugate variables. This extension is useful in some cases to get action-angle variables from the general solution of a given integrable Hamiltonian system. As an example the Delaunay variables are proved to be canonical.

ON THE ONE-DIMENSIONAL SCHRÖDINGER EQUATION WITH A QUASI-PERIODIC POTENTIAL

Nondegeneracy in the Perturbation Theory of Integrable Dynamical Systems

The most general nondegeneracy condition for the existence of invariant tori in nearly integrable and analytic Hamiltonian systems is formulated.

On the existence of invariant curves of twist mappings of an annulus

On optimal estimates for the solutions of linear difference equations on the circle

A linear difference equation arising in the proof of Moser's twist mapping theorem is solved and optimal estimates for the solution are established.

Convergent transformations into a normal form in analytic Hamiltonian systems with two degrees of freedom on the zero energy surface near degenerate elliptic singularities

We study an analytic Hamiltonian system with two degrees of freedom, having the origin as an elliptic singularity. We assume that the full Birkhoff normal form exists and is divisible by its quadratic part, being indefinite. We show that under the Bruno condition and under the restriction to the zero energy surface, a real analytic transformation into a normal form exists. Such a normal form coincides with the restriction of the Birkhoff normal form to the zero energy surface up to an order as large as we want.

On an Inequality for Trigonometric Polynomials In Several Variables

Publisher Summary This chapter presents trigonometric polynomials in n variables. Using the methods of approximation theory, an inequality can be extended to almost periodic functions and to still more general classes of functions as in the case for Bohr's inequality. However, no analogous result exists in the case of two variables. For the solution of problems containing small divisors, the estimate has to be completed by theorems concerning the best approximation of holomorphic functions by trigonometric polynomials in polystrips. The chapter also presents equations to provide an estimate for a differential operator.

On a new proof of Moser's twist mapping theorem

Based on a new idea of the author, a new proof of J. Moser's twist mapping theorem is presented.