On Tonelli periodic orbits with low energy on surfaces

Asselle, Luca; Mazzucchelli, Marco

doi:10.1090/tran/7185

On Tonelli periodic orbits with low energy on surfaces
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by Luca Asselle and Marco Mazzucchelli PDF

Trans. Amer. Math. Soc. 371 (2019), 3001-3048 Request permission

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.

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