Quasi-periodic Swing via Weak KAM Theory

We investigate the dynamics of the quasi-periodic swing equations from the perspective of weak KAM theory. To this end, we firstly study a class of Hamiltonian systems. We obtain that the limit $u$, which derived from convergence of a sequence of functional minimizers, satisfies Hamilton-Jacobi equations in a weak sense. This is the so-called weak KAM solution. Meanwhile, we also get aminimal measures $\mu$. Finally, we derive that the existence of weak KAM solutions for the swing equations.