Invariant Lagrangian graphs, Hamilton-Jacobi equation and action-minizing properties of Tonelli Hamiltonians

In the study of Hamiltonian systems a special role is played by invariant Lagrangian submanifolds. These objects arise quite naturally in many physical and geometric problems and besides sharing a deep relation with the dynamics of the system, they are also closely related to classical and `weak' solutions of the corresponding Hamilton-Jacobi equation(s). When does a Hamiltonian system possess an invariant smooth Lagrangian graph or a family thereof? In this talk I shall discuss how this very interesting (and difficult) question can be approached from different perspectives and describe several results related to the so-called Principle of least Lagrangian action.