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arXiv · 1601.06692

On Tonelli periodic orbits with low energy on surfaces

Luca Asselle·Marco Mazzucchelli
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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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Luca Asselle, Marco Mazzucchelli. 2018-04-06. On Tonelli periodic orbits with low energy on surfaces. https://doi.org/10.1090/tran%2F7185

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