Optimalit\'e systolique infinit\'esimale de l'oscillateur harmonique

TL;DR

This paper investigates the existence of short periodic orbits on energy surfaces of Hamiltonian systems in symplectic geometry, focusing on minimal systolic volume and optimality conditions.

Contribution

It provides new insights into the infinitesimal properties of Hamiltonian energy surfaces and their relation to the existence of short periodic orbits.

Findings

Energy surfaces with certain volume conditions carry periodic orbits with action ≤ π

The study establishes optimality conditions for systolic inequalities in Hamiltonian systems

Results contribute to understanding the minimal action of periodic orbits in symplectic geometry.

Abstract

We study the infinitesimal aspects of the following problem. Let H be a Hamiltonian of \R^{2n} whose energy surface {H=1} encloses a compact starshaped domain of volume equal to that of the unit ball in \R^{2n}. Does the energy surface {H=1} carry a periodic orbit of the Hamiltonian system associated to H with action less than or equal to \pi ?