Tonelli Lagrangian Research Articles

Periodic orbits of Tonelli Lagrangians with prescribed action or period

We show that for every Tonelli Lagrangian there is a real number a 0 a_0 such that for every a > a 0 a>a_0 the Lagrangian has a periodic orbit with action a a . We obtain estimates on the period and energy of such periodic orbits. We also show that such Lagrangian has periodic orbits of every period.

Generalized curvature for the optimal transport problem induced by a Tonelli Lagrangian

We propose a generalized curvature that is motivated by the optimal transport problem on {mathbb {R}}^d with cost induced by a Tonelli Lagrangian L. We show that non-negativity of the generalized curvature implies displacement convexity of the generalized entropy functional on the L-Wasserstein space along C^2 displacement interpolants.

On the $$C^1$$ and $$C^2$$-Convergence to Weak K.A.M. Solutions

We introduce a notion of upper Green regular solutions to the Lax-Oleinik semi-group that is defined on the set of \(C^0\) functions of a closed manifold via a Tonelli Lagrangian. Then we prove some weak \(C^2\) convergence results to such a solution for a large class of approximated solutions as (1) the discounted solution (see [DFIZ16]); (2) the image of a \(C^0\) function by the Lax-Oleinik semi-group; (3) the weak K.A.M. solutions for perturbed cohomology class. This kind of convergence implies the convergence in measure of the second derivatives. Moreover, we provide an example that is not upper Green regular and to which we have \(C^1\) convergence but not convergence in measure of the second derivatives.

Heteroclinic orbits between static classes of time periodic Tonelli Lagrangian systems

<abstract><p>In this paper, we show the existence of heteroclinic orbits between two different static classes of the Aubry set of a Tonelli Lagrangian when the Aubry set has only finite static classes (which is a generic condition in the sense of Mañé) and the Mañé set satisfies certain isolated condition.</p></abstract>

Generic Properties of Mañé’s Set of Exact Magnetic Lagrangians

Let $$M$$ be a closed manifold and $$L$$ an exact magnetic Lagrangian. In this paper we prove that there exists a residual set $$\mathcal{G}$$ of $$H^{1}\left(M;\mathbb{R}\right)$$ such that the property $${\widetilde{\mathcal{M}}}\left(c\right)={\widetilde{\mathcal{A}}}\left(c\right)={\widetilde{\mathcal{N}}}\left(c\right),\forall c\in\mathcal{G},$$ with $${\widetilde{\mathcal{M}}}\left(c\right)$$ supporting a uniquely ergodic measure, is generic in the family of exact magnetic Lagrangians. We also prove that, for a fixed cohomology class $$c$$ , there exists a residual set of exact magnetic Lagrangians such that, when this unique measure is supported on a periodic orbit, this orbit is hyperbolic and its stable and unstable manifolds intersect transversally. This result is a version of an analogous theorem, for Tonelli Lagrangians, proven in [6].

A New Approach for Euler-Lagrange Orbits on Compact Manifolds with Boundary

Consider a compact manifold with boundary, homeomorphic to the N-dimensional disk, and a Tonelli Lagrangian function defined on the tangent bundle. In this paper, we study the multiplicity problem for Euler-Lagrange orbits that satisfy the conormal boundary conditions and that lay on the boundary only in their extreme points. In particular, for suitable values of the energy function and under mild hypotheses, if the Tonelli Lagrangian is reversible then the minimal number of Euler-Lagrange orbits with prescribed energy that satisfies the conormal boundary conditions is N. If L is not reversible, then this number is two.

A Note on Tonelli Lagrangian Systems on $\mathbb{T}^2$ with Positive Topological Entropy on a High Energy Level

In this work we study the dynamical behavior of Tonelli Lagrangian systems defined on the tangent bundle of the torus $\mathbb{T}^2=\mathbb{R}^2/\mathbb{Z}^2$. We prove that the Lagrangian flow restricted to a high energy level $E_{L}^{-1}(c)$ (i.e., $c > c_0(L)$) has positive topological entropy if the flow satisfies the Kupka-Smale property in $E_{L}^{-1}(c)$ (i.e., all closed orbits with energy c are hyperbolic or elliptic and all heteroclinic intersections are transverse on $E_{L}^{-1}(c)$). The proof requires the use of well-known results from Aubry – Mather theory.

Minimal Boundaries in Tonelli Lagrangian Systems

Abstract 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 than the maximal energy of a constant orbit and smaller than or equal to the Mañé critical value of the universal abelian cover, the Lagrangian system admits a minimal boundary, that is, a global minimizer of the Lagrangian action on the space of smooth boundaries of open sets of $M$. We also extend the celebrated graph theorem of Mather in this context: in the tangent bundle $\textrm{T} M$, the union of the supports of all lifted minimal boundaries with a given energy projects injectively to the base $M$. Finally, we prove the existence of action minimizing simple periodic orbits on energies just above the Mañé critical value of the universal abelian cover. This provides in particular a class of nonreversible Finsler metrics on the two-sphere possessing infinitely many closed geodesics.

Waist theorems for Tonelli systems in higher dimensions

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 Lagrangian has no stationary orbit at the Ma\~n\'e critical energy level, there is a waist on every energy level just above the Ma\~n\'e critical value. With a suitable perturbation with a potential, we show that there are infinitely many periodic orbits on every energy level just above the Ma\~n\'e critical value, and on almost every energy level just below. Finally, we prove the Tonelli analogue of a closed geodesics result due to Ballmann-Thorbergsson-Ziller.

Lasry–Lions approximations for discounted Hamilton–Jacobi equations

We study the Lasry–Lions approximation using the kernel determined by the fundamental solution with respect to a time-dependent Tonelli Lagrangian. This approximation process is also applied to the viscosity solutions of the discounted Hamilton–Jacobi equations.

SMOOTH HAMILTON-JACOBI SOLUTIONS FOR THE HOROCYCLE FLOW

In this paper we compute all the smooth solutions to the Hamilton-Jacobi equation associated with the horocycle flow. This can be seen as the Euler-Lagrange flow (restricted to the energy level set $E^{-1}(\frac 12)$) defined by the Tonelli Lagrangian $L:T\mathbb H\rightarrow \mathbb R$ given by (hyperbolic) kinetic energy plus the standard magnetic potential. The method we use is to look at Lagrangian graphs that are contained in the level set $\{H=\frac 12\}$, where $H:T^*\mathbb H\rightarrow \mathbb R$ denotes the Hamiltonian dual to $L$.

The Lax-Oleinik semigroup on graphs

We consider Tonelli Lagrangians on a graph, define weak KAM solutions, which happen to be the fixed points of the Lax-Oleinik semi-group, and identify their uniqueness set as the Aubry set, giving a representation formula. Our main result is the long time convergence of the Lax Oleinik semi-group. It follows that weak KAM solutions are viscosity solutions of the Hamilton-Jacobi equation [ 3 , 4 ], and in the case of Hamiltonians called of eikonal type in [ 3 ], we prove that the converse holds.

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 levels, these systems may have only finitely many periodic orbits. Our main result asserts that almost all energy levels in a precisely characterized intermediate range ( e 0 , e 1 ) ${(e_{0},e_{1})}$ possess infinitely many periodic orbits. Such a range of energies is non-empty, for instance, in the physically relevant case where the Tonelli Lagrangian is a kinetic energy and the magnetic form is oscillating (in which case, e 0 = 0 ${e_{0}=0}$ is the minimal energy of the system).

On the existence of Euler–Lagrange orbits satisfying the conormal boundary conditions

Let (M,g) be a closed connected Riemannian manifold, L:TM→R be a Tonelli Lagrangian. Given two non-empty closed submanifolds Q0,Q1⊆M and a real number k, we study the existence of Euler–Lagrange orbits with energy k connecting Q0 to Q1 and satisfying suitable boundary conditions, known as conormal boundary conditions. We introduce the Mañé critical value which is relevant for this problem and discuss existence results for supercritical and subcritical energies. We also provide counterexamples showing that all the results are sharp.

Global minimizers for Tonelli Lagrangians on the 2-torus

We study action-minimizing orbits in Tonelli Lagrangian systems on the 2-torus on fixed energy levels above Mañé's strict critical value. Our work generalizes the results of Morse, Hedlund and Bangert on minimal geodesics in Riemannian 2-tori. The techniques in the proofs involve classical variational ones, as well as the theories of Mather, Mañé and Fathi, which allow the step from reversible to non-reversible dynamics.

Invariant tori and topological entropy in Tonelli Lagrangian systems on the 2-torus

We study the Euler–Lagrange flow of a Tonelli Lagrangian on the 2-torus$\mathbb{T}^{2}$at a fixed energy level${\mathcal{E}}\subset T\mathbb{T}^{2}$strictly above Mañé’s strict critical value. We prove that, if for some rational direction${\it\zeta}\in S^{1}$there is no invariant graph${\mathcal{T}}\subset {\mathcal{E}}$over$\mathbb{T}^{2}$for the Euler–Lagrange flow with the property that all orbits on${\mathcal{T}}$have an asymptotic direction equal to${\it\zeta}$, then there are chaotic dynamics in${\mathcal{E}}$. This implies that, if the topological entropy of the Euler–Lagrange flow in${\mathcal{E}}$vanishes, then in${\mathcal{E}}$there are invariant graphs for all asymptotic directions${\it\zeta}\in S^{1}$and integrable-like behavior on a large scale.

弱KAM理论和Hamilton-Jacobi方程

Mather theory studies the existence of various kinds of the action minimizing sets and the connecting orbits between these sets in higher-dimensional positive Lagrangian system. The work about the connecting orbits plays an important role in the study of Arnold diffusion. By studying the dynamical behavior of the action-minimizing curves for Tonelli Lagrangian systems, weak KAM theory founded by A. Fathi bridges Mather theory and the PDE methods concerning the associated Hamilton-Jacobi equation. However, because the convergence of the Lax-Oleinik semigroup which is critical in KAM theory does not hold in time-periodic Lagrangian system, the preliminary work about weak KAM theory focused on autonomous system. By introducing a new kind of Lax-Oleinik type operator, it is possible for us to build weak KAM theory in the time-periodic Lagrangian system and generalize the theory to more general Hamilton-Jacobi equation. In this paper, we introduce the basic knowledge of weak KAM theory and its latest development.

Some Results on Weak KAM Theory for Time-periodic Tonelli Lagrangian Systems

Abstract This paper contributes several results on weak KAM theory for time-periodic Tonelli Lagrangian systems. Wang and Yan [Commun. Math. Phys. 309 (2012), 663-691] introduced a new kind of Lax-Oleinik type operator associated with any time-periodic Tonelli Lagrangian. Firstly, using the new operator we give an equivalent definition of the backward weak KAM solution. Then we prove a result on the asymptotic behavior of the new operators with an arbitrary continuous function as initial condition, by taking advantage of the definition mentioned above. Finally, for a specific class of time-periodic Tonelli Lagrangians, we discuss the rate of convergence of the new operators.

Aubry sets vs. Mather sets in two degrees of freedom

Let L be an autonomous Tonelli Lagrangian on a closed manifold of dimension two. We prove that for a large set of cohomology classes, the Aubry set consists of periodic orbits.

The Lagrangian Conley conjecture

We prove a Lagrangian analogue of the Conley conjecture: given a 1-periodic Tonelli Lagrangian with global flow on a closed configuration space, the associated Euler–Lagrange system has infinitely many periodic solutions. More precisely, we show that there exist infinitely many contractible integer periodic solutions with a priori bounded mean action and either infinitely many of them are 1-periodic or they have unbounded period.