Multidimensional Euler-Poincaré equations

Abstract

This work is devoted to presenting a summary of results developed mainly by the first author, T. S. Ratiu and S. Shkoller in their previous work [Proc. Amer. Math. Soc. 128 (2000), no. 7, 2155–2164;]. The main results concern the reduction of a Lagrangian field theory under a group of symmetries, obtaining the analog of the Euler-Poincaré equations, which are also proved to be equivalent to a Noether conservation law given by the symmetry. Furthermore, the compatibility condition needed for obtaining solutions of the original variational problem starting from the solutions of the reduced system is also stated. A final example is given. The paper is written in geometrical language.