Lagrangian Reduction by Stages

Abstract

This paper studies the geometry of the reduction of Lagrangian sys-tems with symmetry in a way that allows the reduction process to berepeated; that is, it develops a context for Lagrangian reduction bystages. The Lagrangian reduction procedure focusses on the geometryof variational structures and how to reduce them. This philosophy iswell known for the classical cases, such as those of Routh (where thesymmetry group is Abelian) and the Euler{Poincaré equations (for thecase in which the con guration space is a Lie group).The context established for this theory is a Lagrangian analogue ofthe bundle picture on the Hamiltonian side. In this picture, a cate-gory is developed that includes, as a special case, the realization of thequotient of a tangent bundle as the Whitney sum of the tangent of thequotient bundle with the associated adjoint bundle. The elements of this new category, called the Lagrange-Poincaré category, have enoughgeometric structure so that the category is stable under the procedureof Lagrangian reduction. Thus, reduction may be repeated, giving thedesired context for reduction by stages.We also give an intrinsic and geometric way of writing the reducedequations, called the Lagrange-Poincaré equations, using covariant de-rivatives and connections. In addition, the context includes the inter-pretation of cocycles as curvatures of connections and is general enoughto include interesting situations involving both semidirect products andcentral extensions. Examples are given to illustrate the general theory.