From Tools in Symplectic and Poisson Geometry to J.-M. Souriau’s Theories of Statistical Mechanics and Thermodynamics

Entropy

Charles-Michel Marle¹ 

[1] Institut de Mathématiques de Jussieu, Université Pierre et Marie Curie, 4, Place Jussieu, 75252 Paris Cedex 05, France;

关键词: Lagrangian formalism;    Hamiltonian formalism;    symplectic manifolds;    Poisson structures;    symmetry groups;    momentum maps;    thermodynamic equilibria;    generalized Gibbs states;   

DOI  :  10.3390/e18100370

来源: DOAJ

【 摘 要 】

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 an 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 fully discussed and examples of applications in physics are given.

【 授权许可】

Unknown