Quasi-periodic swing via weak KAM theory

Abstract

Our primary focus is on the study of the dynamics of quasi-periodic swing equations from the weak KAM point of view. To achieve this, we initially explore a class of quasi-periodic Hamiltonian systems. We discover that a limit function, derived from the convergence of a sequence of functional minimizers, satisfies the Hamilton–Jacobi equations in the context of minimal measures. This is the so-called weak KAM solution. Subsequently, we establish the existence of invariant torus for the swing equation in a weak sense. Lastly, we discuss certain properties of the weak KAM solution for a one-dimensional periodic swing equation.