On Tonelli periodic orbits with low energy on surfaces

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...

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Bibliographic Details
Published in:Transactions of the American Mathematical Society 2019-03, Vol.371 (5), p.3001-3048
Main Authors: Asselle, Luca, Mazzucchelli, Marco
Format: Article
Language:English
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Summary: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.
ISSN:0002-9947
1088-6850
DOI:10.1090/tran/7185