Constructive approaches to QP-time-dependent KAM theory for Lagrangian tori in Hamiltonian systems

Abstract

In this paper, we prove a KAM theorem in a-posteriori format, using the parameterization method to look invariant tori in non-autonomous Hamiltonian systems with n degrees of freedom that depend periodically or quasi-periodically (QP) on time, with pound external frequencies. Such a system is described by a Hamiltonian function in the 2n-dimensional phase space, & Mscr;, that depends also on pound angles, phi E T pound. We take advantage of the fibered structure of the extended phase space & Mscr; x T pound. As a result of our approach, the parameterization of tori requires the last pound variables, to be precise phi, while the first 2n components are determined by an invariance equation. This reduction decreases the dimension of the problem where the unknown is a parameterization from 2(n + ) pound to 2n. We employ a quasi-Newton method, in order to prove the KAM theorem. This iterative method begins with an initial parameterization of an approximately invariant torus, meaning it approximately satisfies the invariance equation. The approximation is refined by applying corrections that reduce quadratically the invariance equation error. This process converges to a torus in a complex strip of size rho er, provided suitable Diophantine (gamma, tau) conditions and a non-degeneracy condition on the torsion are met. Given the nature of the proof, this provides a numerical method that can be effectively implemented on a computer, the details are given in the companion paper [9]. This approach leverages precision and efficiency to compute invariant tori.