Searcharxiv⌕ Search

arXiv subjects

Luca Asselle

Publications and source records attributed to Luca Asselle.

25 records · Page 2Linked to original sources
arXiv2016

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

Let $(\mathbb T^2,g)$ be a Riemannian two-torus and let $σ$ be an oscillating $2$-form on $\mathbb T^2$. We show that for almost every small positive number $k$ the magnetic flow of the pair $(g,σ)$ has infinitely many periodic orbits with energy $k$. This result complements the analogous statement for closed surfaces of genus at least $2$ [Asselle and Benedetti, Calc. Var. Partial Differential Equations, 2015] and at the same time extends the main theorem in [Abbondandolo, Macarini, Mazzucchelli, and Paternain, J. Eur. Math. Soc. (JEMS), to appear] to the non-exact oscillating case.

math.DS↗
arXiv2016

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

Let $(M,g)$ be a closed Riemannian manifold, $L: TM\rightarrow \mathbb R$ be a Tonelli Lagrangian. Given two closed submanifolds $Q_0$ and $Q_1$ of $M$ and a real number $k$, we study the existence of Euler-Lagrange orbits with energy $k$ connecting $Q_0$ to $Q_1$ and satisfying the 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.

math.DS↗
arXiv2016

The multiplicity problem for periodic orbits of magnetic flows on the 2-sphere

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 $(e_0,e_1)$ 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, $e_0=0$ is the minimal energy of the system).

math.DS↗
arXiv2016

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 $L$.

math.DS↗
arXiv2015

On the existence of orbits satisfying periodic or conormal boundary conditions for Euler-Lagrange flows

Let $(M,g)$ be a closed Riemannian manifold and $L:TM\rightarrow \mathbb R$ be a Tonelli Lagrangian. In this thesis we study the existence of orbits of the Euler-Lagrange flow associated with $L$ satisfying suitable boundary conditions. We first look for orbits connecting two given closed submanifolds of $M$ satisfying the conormal boundary conditions: We introduce the Mañé critical value that is relevant for the problem and prove existence results for supercritical and subcritical energies; we also complement these with counterexamples, thus showing the sharpness of our results. We then move to the problem of finding periodic orbits: We provide an existence result of periodic orbits for non-aspherical manifolds generalizing the Lusternik-Fet Theorem, and a multiplicity result in case the configuration space is the 2-torus.

math.DS↗
arXiv2015

The Lusternik-Fet theorem for autonomous Tonelli Hamiltonian systems on twisted cotangent bundles

Let $M$ be a closed manifold and consider the Hamiltonian flow associated to an autonomous Tonelli Hamiltonian $H:T^*M\rightarrow \mathbb R$ and a twisted symplectic form. In this paper we study the existence of contractible periodic orbits for such a flow. Our main result asserts that if $M$ is not aspherical, then contractible periodic orbits exist for almost all energies above the maximum critical value of $H$.

math.SG↗