  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 Poincar\'e disk and the Poincar\'e
half-plane.
