Geometry has proven to be a powerful language in which the formulation of variational principles reveals their basic properties. From Mechanics to Field theories, variational problems described in the framework of manifolds, bundles and Lie group actions provide a convenient setting enjoying a long an active scientific production. In particular, one of the main cornerstones in this geometric formulation is the idea of reduction. When a Lagrangian is invariant with respect to certain group of symmetries, this idea consists of the simplification of the configuration space of the variational problem. However, there are many interesting situations where the group of symmetries is composed by different subgroups of different nature (for example, rigid bodies with internal rotors, fluids, ...). In this case, it is convenient to reduce first by a subgroup and then by the rest of the symmetries. This is known as Reduction by Stages. Unfortunately, once a single reduction is performed, the configuration space is not longer a tangent (or jet) space, and the standard variational calculus must be redesigned. In this talk we will gently review the reduction by stages scheme and introduce the so-called category of Lagrange-Poincaré configurations spaces, first Mechanics and then in Field Theories. Some examples and applications will be provided.
Meeting ID: 814 591 7621 Passcode: 147983