# On Tonelli periodic orbits with low energy on surfaces

@article{Asselle2016OnTP, title={On Tonelli periodic orbits with low energy on surfaces}, author={Luca Asselle and Marco Mazzucchelli}, journal={Transactions of the American Mathematical Society}, year={2016} }

<p>We prove that on a closed surface, a Lagrangian system defined by a Tonelli Lagrangian <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L">
<mml:semantics>
<mml:mi>L</mml:mi>
<mml:annotation encoding="application/x-tex">L</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> possesses a periodic orbit that is a local minimizer of the free-period action functional on every energy level belonging to the…

## 13 Citations

### Minimal Boundaries in Tonelli Lagrangian Systems

- MathematicsInternational Mathematics Research Notices
- 2019

We prove several new results concerning action minimizing periodic orbits of Tonelli Lagrangian systems on an oriented closed surface $M$. More specifically, we show that for every energy larger…

### Infinitely many periodic orbits just above the Mañé critical value on the 2-sphere

- Mathematics
- 2017

We introduce a new critical value $c_\infty(L)$ for Tonelli Lagrangians $L$ on the tangent bundle of the 2-sphere without minimizing measures supported on a point. We show that $c_\infty(L)$ is…

### The Multiplicity Problem for Periodic Orbits of Magnetic Flows on the 2-Sphere

- Physics, Mathematics
- 2016

Abstract We consider magnetic Tonelli Hamiltonian systems on the cotangent bundle of the 2-sphere, where the magnetic form is not necessarily exact. It is known that, on very low and on high energy…

### Lecture notes on closed orbits for twisted autonomous Tonelli Lagrangian flows

- Mathematics, Physics
- 2016

These notes were prepared in occasion of a mini-course given by the author at the “CIMPA Research School Hamiltonian and Lagrangian Dynamics” (10–19 March 2015 Salto, Uruguay). The talks were meant…

### Waist theorems for Tonelli systems in higher dimensions

- Physics, Mathematicsmanuscripta mathematica
- 2019

We study the periodic orbits problem on energy levels of Tonelli Lagrangian systems over configuration spaces of arbitrary dimension. We show that, when the fundamental group is finite and the…

### Periodic orbits in oscillating magnetic fields on $\mathbb T^2$

- Mathematics
- 2015

Let $(M,g)$ be a closed connected orientable Riemannian surface and let $\sigma$ be a 2-form on $M$ such that its density with respect to the area form induced by $g$ attains both positive and…

### A new approach to the existence of closed magnetic geodesics via symplectic reduction

- Mathematics
- 2016

Let $(M,g)$ be a closed Riemannian manifold and $\sigma$ be a closed 2-form on $M$ representing an integer cohomology class. In this paper, using symplectic reduction, we show how the problem of…

### Non-resonant circles for strong magnetic fields on surfaces

- Physics, MathematicsAnnales Henri Lebesgue
- 2022

. — We study non-resonant circles for strong magnetic ﬁelds on a closed, connected, oriented surface and show how these can be used to prove the existence of trapping regions and of periodic magnetic…

### Minimax Periodic Orbits of Convex Lagrangian Systems on Complete Riemannian Manifolds

- MathematicsThe Journal of Geometric Analysis
- 2022

In this paper we study the existence of periodic orbits with prescribed energy levels of convex Lagrangian systems on complete Riemannian manifolds. We extend the existence results of Contreras by…

### On the existence of closed magnetic geodesics via symplectic reduction

- Materials ScienceJournal of Fixed Point Theory and Applications
- 2018

Let (M, g) be a closed Riemannian manifold and σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy}…

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Abstract We consider magnetic Tonelli Hamiltonian systems on the cotangent bundle of the 2-sphere, where the magnetic form is not necessarily exact. It is known that, on very low and on high energy…

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