The second Birkhoff theorem for optical Hamiltonian systems

Source abstract

Consider a smooth Hamiltonian function on the cotangent bundle of the n n -dimensional torus such that its restriction on every fiber is strictly convex. Let L L be a Lagrange invariant torus of the Hamiltonian flow which is homologous to the zero section. We show that, under some assumptions, L L is a smooth section of the cotangent bundle.