EFFICIENT CALCULATION OF ACTIONS

We present a method to calculate numerically the action variables of a completely integrable Hamiltonian system with N degrees of freedom. I t is a construction of the Liouville-Arnol'd theorem for the existence of tori in phase space. By introducing a metric on phase space the pr oblem of finding N independent irreducible paths on a given torus is t urned into the problem of finding the lattice of zeros of an N-periodi c function. This function is constructed using the flows of all consta nts of motion. Using the fact that neighbouring tori and their irreduc ible paths are related by some continuous deformation, a continuation method is constructed which allows a systematic scan of the actions. F or N = 2 we use a Poincare surface of section to define paths which cr oss neighbouring tori. Close to isolated periodic orbits the generator s are either constructed explicitly or their asymptotic behaviour is g iven. As an example, the energy surface in the space of action variabl es of a Hamiltonian showing resonances is calculated.