# On Tonelli periodic orbits with low energy on surfaces

@article{Asselle2018OnTP, title={On Tonelli periodic orbits with low energy on surfaces}, author={Luca Asselle and Marco Mazzucchelli}, journal={Transactions of the American Mathematical Society}, year={2018} }

<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…

## 12 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…

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

- Mathematics
- 2016

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

### 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…

### 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 geodesic flows with symmetries and closed magnetic geodesics on orbifolds

- MathematicsErgodic Theory and Dynamical Systems
- 2020

Let $Q$ be a closed manifold admitting a locally free action of a compact Lie group $G$. In this paper, we study the properties of geodesic flows on $Q$ given by suitable G-invariant Riemannian…

## References

SHOWING 1-10 OF 40 REFERENCES

### On the periodic motions of a charged particle in an oscillating magnetic field on the two-torus

- Mathematics
- 2015

Let $$({\mathbb {T}}^2,g)$$(T2,g) be a Riemannian two-torus and let $$\sigma $$σ be an oscillating 2-form on $${\mathbb {T}}^2$$T2. We show that for almost every small positive number k the magnetic…

### Lagrangian Graphs, Minimizing Measures and Mañé's Critical Values

- Mathematics
- 1998

Abstract. Let
$\Bbb L$ be a convex superlinear Lagrangian on a closed connected manifold N. We consider critical values of Lagrangians as defined by R. Mañé in [M3]. We show that the critical value…

### 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…

### Periodic orbits of magnetic flows for weakly exact unbounded forms and for spherical manifolds

- Mathematics
- 2014

We show that for weakly exact magnetic flows with infinite Ma\~n\'e critical value the action functional satisfies the Palais-Smale condition on the space of contractible loops with period bounded…

### Closed extremals on two-dimensional manifolds

- Mathematics
- 1992

CONTENTSIntroductionChapter I. Closed geodesies on two-dimensional manifolds1. Geodesic lines: definition and main properties2. The minimax principle and the Birkhoff theorem3. The…

### Lagrangian flows: The dynamics of globally minimizing orbits-II

- Mathematics
- 1997

Define the critical levelc(L) of a convex superlinear LagragianL as the infimum of thek ∈ ℝsuch that the LagragianL+k has minimizers with fixed endpoints and free time interval. We provide proofs for…

### 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…

### Action-minimizing Methods in Hamiltonian Dynamics (MN-50): An Introduction to Aubry-Mather Theory

- Mathematics
- 2015

John Mather’s seminal works in Hamiltonian dynamics represent some of the most important contributions to our understanding of the complex balance between stable and unstable motions in classical…

### An Introduction to Riemann-Finsler Geometry

- Mathematics
- 2000

One Finsler Manifolds and Their Curvature.- 1 Finsler Manifolds and the Fundamentals of Minkowski Norms.- 1.0 Physical Motivations.- 1.1 Finsler Structures: Definitions and Conventions.- 1.2 Two…

### Residual finiteness of surface groups

- Mathematics
- 1972

It is known [2] that free groups, and more generally fundamental groups of 2-manifolds [1], are residually finite. We give here an elementary proof of these facts. THEOREM. Let F be a (possibly…