  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.
