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… 

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References

SHOWING 1-10 OF 40 REFERENCES

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

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

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

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

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

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

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 $\mathbb T^2$

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

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

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

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