It is shown that self-locomotion is possible for a body in Euclidian space, provided its dynamics corresponds to a non-quadratic Hamiltonian, and that the body contains at least 3 particles. The efficiency of the driver of such a system is defined. The existence of an optimal (most efficient) driver is proved.


We study the zero-temperature limit for Gibbs measures associated to Frenkel–Kontorova models on ${\left({ℝ}^d\right)}^{\mathbb{Z}}/{\mathbb{Z}}^d$. We prove that equilibrium states concentrate on configurations of minimal energy, and, in addition, must satisfy a variational principle involving metric entropy and Lyapunov exponents, a bit like in the Ruelle–Pesin inequality. Then we transpose the result to certain continuous-time stationary stochastic processes associated to the viscous Hamilton–Jacobi equation. As the viscosity vanishes, the...

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 $\pi$ ?