Covariant Poisson Reduction: First Steps

Marco Castrillon Lopez, Mathematics, Universidad Complutense de Madrid

The Theory of Symplectic and Poisson reduction in Mechanics opened an important source of interesting problems and questions both in Mathematics and in Physics. Similarly, the Theory of Lagrangian reduction (Euler-Poincaré, Lagrange-Poincaré) has shown to be a powerful tool in many different aspects of Geometric Mechanics. One can ask if all these ideas can be applied to covariant variational problems in Field Theory. The first attempts done in this direction were in the Lagrangian side with the covariant Euler-Poincaré reduction for principal bundles. We now explore the analog of Lie-Poisson reduction in principal bundles. The goals of the talk will be the introduction of the covariant Poisson formulation for field problems and the reduction process when the configuration bundle is a $G$ principal bundle and the Hamiltonian is $G$ invariant. The implications and the future work suggested by these results will be also commented.