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Entropy 2016, 18(10), 370; doi:10.3390/e18100370

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

Institut de Mathématiques de Jussieu, Université Pierre et Marie Curie, 4, Place Jussieu, 75252 Paris Cedex 05, France
In memory of Jean-Marie Souriau (1922–2012).
Academic Editors: Frédéric Barbaresco and Frank Nielsen
Received: 28 July 2016 / Revised: 30 September 2016 / Accepted: 5 October 2016 / Published: 19 October 2016
(This article belongs to the Special Issue Differential Geometrical Theory of Statistics)
View Full-Text   |   Download PDF [450 KB, uploaded 19 October 2016]

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 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. View Full-Text
Keywords: Lagrangian formalism; Hamiltonian formalism; symplectic manifolds; Poisson structures; symmetry groups; momentum maps; thermodynamic equilibria; generalized Gibbs states Lagrangian formalism; Hamiltonian formalism; symplectic manifolds; Poisson structures; symmetry groups; momentum maps; thermodynamic equilibria; generalized Gibbs states
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Marle, C.-M. From Tools in Symplectic and Poisson Geometry to J.-M. Souriau’s Theories of Statistical Mechanics and Thermodynamics. Entropy 2016, 18, 370.

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