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

TL;DR

This paper explores symplectic and Poisson geometry tools and their applications in geometric mechanics, statistical mechanics, and thermodynamics, emphasizing Souriau's theories and the generalization of thermodynamic equilibrium.

Contribution

It introduces geometric tools and concepts, such as manifold of motions and generalized equilibrium, to connect symplectic geometry with statistical mechanics and thermodynamics.

Findings

Application of symplectic and Poisson geometry in physics

Generalization of thermodynamic equilibrium concepts

Examples of geometric methods in physical systems

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.