From tools in symplectic and Poisson geometry to Souriau's theories of statistical mechanics and thermodynamics

Abstract: I present in this paper some tools in Symplectic and Poisson Geometry in view of their applications in Geometric mechanics and Mathematical Physics. After a short discussion of the Lagrangian and Hamiltonian formalisms, including the use of symmetry groups, and a presentation of the Tulczyjew's isomorphisms (which explain some aspects of the relations between these formalisms), I explain the concept of manifold of motions of a mechanical system and its use, due to J.-M. Souriau, in Statistical Mechanics and Thermodynamics. The generalization of the notion of thermodynamic equilibrium in which the one-dimensional group of time translations is replaced by a multi-dimensional, maybe non-commutative Lie group, is discussed and examples of applications in Physics are given.