On Gibbs states of mechanical systems with symmetries

Abstract

Gibbs states for the Hamiltonian action of a Lie group on a symplectic manifold were studied, and their possible applications in Physics and Cosmology were considered, by the French mathematician and physicist Jean-Marie Souriau. They are presented here with detailed proofs of all the stated results. Using an adaptation of the cross product for pseudo-Euclidean three-dimensional vector spaces, we present several examples of such Gibbs states, together with the associated thermodynamic functions, for various two-dimensional symplectic manifolds, including the pseudo-spheres, the Poincaré disk and the Poincaré half-plane.

arXiv comment59 pages, preprint. This version differs from the previoos one by corrections of several typographical errors and addition of the expressions of the Liouville measure for the Poincaré disk and the Poincaré half-plane