  In this paper, we show the existence of non contractible periodic orbits in
Hamiltonian systems defined on $T^*\T^n$ separating two Lagrangian tori under
certain cone assumption. Our result answers a question of Polterovich in
\cite{P} in a sharp way. As an application, we find periodic orbits of almost
all the homotopy types on a dense set of energy level in Lorentzian type
mechanical Hamiltonian systems defined on $T^*\T^2$. This solves a problem of
Arnold in \cite{A}.
