A KAM theorem without action-angle variables for elliptic lower dimensional tori

Abstract

We study elliptic lower dimensional invariant tori of Hamiltonian systems via parametrizations. The method is based on solving iteratively the functional equations that stand for invariance and reducibility. In contrast with classical methods, we do not assume that the system is close to an integrable one nor that it is written in action-angle variables. We only require an approximation of an invariant torus with a fixed vector of basic frequencies and a basis along the torus that approximately reduces the normal variational equations to constant coefficients. We want to highlight that this approach presents many advantages compared with methods which are built in terms of canonical transformations, e.g., it produces simpler and more constructive proofs that lead to more efficient numerical algorithms for the computation of these objects. Such numerical algorithms are suitable to be adapted in order to perform computer assisted proofs.