On Tonelli periodic orbits with low energy on surfaces

Abstract

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.

48 pages, 8 figures. Final version. To appear in Trans. Amer. Math. Soc

References in corpus (5)

• The Lusternik-Fet theorem for autonomous Tonelli Hamiltonian systems on twisted cotangent bundles
• The multiplicity problem for periodic orbits of magnetic flows on the 2-sphere
• On the periodic motions of a charged particle in an oscillating magnetic field on the two-torus
• The Morse index of Chaperon's generating families
• Isometry-invariant geodesics and the fundamental group, II

Cited by in corpus (2)

• Minimal boundaries in Tonelli Lagrangian systems
• Waist theorems for Tonelli systems in higher dimensions