Related papers: On Tonelli periodic orbits with low energy on surf…

This paper gives two results that show that the dynamics of a time-periodic Lagrangian system on a hyperbolic manifold are at least as complicated as the geodesic flow of a hyperbolic metric. Given a hyperbolic geodesic in the Poincar\'e…

By a classical theorem of Chekanov, the displacement energy, $e$, of a Lagrangian submanifold is bounded from below by the minimal area of pseudo-holomorphic disks with boundary on the Lagrangian, $\hbar$. We compute $e$ and $\hbar$ for…

We establish regularity results for critical points to energies of immersed surfaces depending on the first and the second fundamental form exclusively. These results hold for a large class of intrinsic elliptic Lagrangians which are…

We consider a periodic problem for the motion of a charged particle in a magnetic field. Introducing a notion of Ricci curvature for such Lagrangian systems and using the methods of the calculus of variations in the large, we prove the…

This paper is about the existence of periodic orbits near an equilibrium point of a two-degree-of-freedom Hamiltonian system. The equilibrium is supposed to be a nondegenerate minimum of the Hamiltonian. Every sphere-like component of the…

We study the existence of non-collision periodic solutions with Newtonian potentials for the following planar restricted 4-body problems: Assume that the given positive masses $m_{1},m_{2},m_{3}$ in a Lagrange configuration move in circular…

Complementary to existing applications of Lagrangian descriptors as an exploratory method, we use Lagrangian descriptors to find invariant manifolds in a system where some invariant structures have already been identified. In this case we…

We prove the existence of periodic bounce orbits of prescribed energy on an open bounded domain in Euclidean space. We derive explicit bounds on the period and the number of bounce points.

In the circular restricted three-body problem, low energy transit orbits are revealed by linearizing the governing differential equations about the collinear Lagrange points. This procedure fails when time-periodic perturbations are…

Let $S$ be a closed surface of genus $g\geq 1$, furnished with an area form $\omega$. We show that there exists an open and dense set ${\mathcal O_r}$ of the space of Hamiltonian diffeomorphisms of class $C^r$, $1\leq r\leq\infty$, endowed…

We first take into account variational problems with periodic boundary conditions, and briefly recall some sufficient conditions for a periodic solution of the Euler-Lagrange equation to be either a directional, a weak, or a strong local…

We will show that the period $T$ of a closed orbit of the planar circular restricted three-body problem (viewed on rotating coordinates) depends on the region it encloses. Roughly speaking, we show that, $2 T=k\pi+\int_\Omega g$ where $k$…

By folding an autonomous system of rational equations in the plane to a scalar difference equation, we show that the rational system has coexisting periodic orbits of all possible periods as well as stable aperiodic orbits for certain…

We study two-degree-of-freedom Hamiltonian systems. Let us assume that the zero energy level of a real-analytic Hamiltonian function $H:\mathbb{R}^4 \to \mathbb{R}$ contains a saddle-center equilibrium point lying in a strictly convex…

Lagrangian multiforms provide a variational framework to describe integrable hierarchies. The case of Lagrangian $1$-forms covers finite-dimensional integrable systems. We use the theory of Lie dialgebras introduced by Semenov-Tian-Shansky…

In this paper we study the relationship between the strict locally minimizing orbits for time dependent lagrangian systems and hyperbolicity properties of the corresponding lagrangian flow.

We study autonomous Tonelli Lagrangians on closed surfaces. We aim to clarify the relationship between the Aubry set and the Mather set, when the latter consists of periodic orbits which are not fixed points. Our main result says that in…

We consider energies on a periodic set ${\mathcal L}$ of ${\mathbb R}^d$ of the form $\sum_{i,j\in{\mathcal L}} a^\varepsilon_{ij}|u_i-u_j|$, defined on spin functions $u_i\in\{0,1\}$, and we suppose that the typical range of the…

We establish a general criterion for the existence of finite energy foliations on contact three-manifolds with boundary, imposing strong global constraints on the associated Reeb flows. Our main abstract result shows that a configuration of…

We study a class of maps having the Collatz function (famously related to the Collatz Conjecture) as an example, under the topological and ergodic perspectives, including an approach with thermodynamic formalism. By introducing a key…