Efficient Calculation of Actions

We present a method to numerically calculate the action variables of a completely integrable Hamiltonian system with N degrees of freedom. It is a constructification of the Liouville-Arnol’d theorem for the existence of tori in phase space. By introducing a metric on phase space the problem of finding N independent irreducible paths on a given torus is turned into the problem of finding the lattice of zeroes of an N -periodic function. This function is constructed using the flows of all constants of motion. For N = 2 we use a Poincaré surface of section to scan all tori with a continuation method. As an example the energy surface in the space of action variables of a Hamiltonian showing resonances is calculated.