From Tools in Symplectic and Poisson Geometry to Souriau's Theories of Statistical Mechanics and Thermodynamics
Résumé
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 Ther-
modynamics. 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.
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