  We prove that, on a closed surface, a Lagrangian system defined by a Tonelli
Lagrangian $L$ possesses a periodic orbit that is a local minimizer of the
free-period action functional on every energy level belonging to the low range
of energies $(e_0(L),c_{\mathrm{u}}(L))$. We also prove that almost every
energy level in $(e_0(L),c_{\mathrm{u}}(L))$ possesses infinitely many periodic
orbits. These statements extend two results, respectively due to Taimanov and
Abbondandolo-Macarini-Mazzucchelli-Paternain, valid for the special case of
electromagnetic Lagrangians.
