We prove that on a closed surface, a Lagrangian system defined by a Tonelli Lagrangian L L possesses a periodic orbit that is a local minimizer of the free-period action… Click to show full abstract
We prove that on a closed surface, a Lagrangian system defined by a Tonelli Lagrangian LL 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 (e0(L),cu(L))(e_0(L),c_{\mathrm {u}}(L)). We also prove that almost every energy level in (e0(L),cu(L))(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.
