Optimalité systolique infinitésimale de l’oscillateur harmonique

Paiva, J.C. Álvarez; Balacheff, Florent

Séminaire de théorie spectrale et géométrie, Tome 27 (2008-2009), pp. 11-16.

Nous étudions les aspects infinitésimaux du problème suivant. Soit $H$ un hamiltonien de ${ℝ}^{2n}$ dont la surface d’énergie $\{H=1\}$ borde un domaine compact et étoilé de volume identique à celui de la boule unité de ${ℝ}^{2n}$. La surface d’énergie $\{H=1\}$ contient-elle une orbite périodique du système hamiltonien

$$\left\{\begin{array}{cc}\dot{q}\hfill & =\frac{\partial H}{\partial p}\hfill \\ \dot{p}\hfill & =-\frac{\partial H}{\partial q}\hfill \end{array}\right.$$

dont l’action soit au plus $\pi$ ?

We study the infinitesimal aspects of the following problem. Let $H$ be a Hamiltonian on ${ℝ}^{2n}$ 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

$$\left\{\begin{array}{cc}\dot{q}\hfill & =\frac{\partial H}{\partial p}\hfill \\ \dot{p}\hfill & =-\frac{\partial H}{\partial q}\hfill \end{array}\right.$$

with action less than or equal to $\pi$ ?

Classification : 37J40, 37J50, 53D10
Mots clés : Forme normale, oscillateur harmonique, volume systolique, système hamiltonien.

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