Kolev, Boris (CNRS and Aix-Marseille University, CMI, 39 rue F. Joliot-Curie, 13453 Marseille Cedex 13, France)
I discuss the Euler-Poincar´e equation as described in Poincaré’s original paper. I consider, then, Lagrangian systems with lot of symmetries, and for which this equation simplifies. Typical examples are furnished by semi-invariant Riemannian metrics on Lie groups. In that case, the Euler-Poincaré equation reduces to a generalization (to an arbitrary Lie group) of the Euler equation for the rigid body. I present several applications of this formalism in mathematical physics.