Conservation laws as part of Lagrangian reduction. Application to image evolution

Abstract

This note collects a series of results into a single formulation. Namely, it is proved that, when reduction is performed to a symmetric Lagrangian, the reduced variational equations can be split into two parts, one of them is exactly the Noether theorem. Conservation laws enter into reduction as part of the new variational equations. We give a short description of this situation in the case of evolution of circles in the plane.