The most concrete version of the question is :
A (necessarily) invariant Lagrangian torus $L$ on the unit cotangent of a Riemannian metric on the two-torus carries a periodic orbit with period $T$. Is it true that every orbit on $L$ is periodic of period $T$?
How about if we also assume that $L$ is a graph?
A more fanciful version of the question is whether there are any obstructions to a flow on a torus $L$ to be realized as the restriction of a Hamiltonian flow to an invariant Lagrangian torus.
