Momentum Maps for Lattice EPDiff

Abstract

This chapter discusses the application of variational particle mesh (VPM) method to the Euler–Poincare (EP) equation on the diffeomorphisms (EPDiff equation). It introduces a constrained variational principle for the method and provides discrete EP formulae on the Eulerian grid resulting from the variational principle, which show that the grid velocities and momenta satisfy the EPDiff equations in Eulerian form. It also discusses left- and right-actions of velocity vector fields on the Lagrangian particles and obtains corresponding momentum maps. The left-action, when restricted to the finite space of velocity fields used in the method, gives rise to a momentum map, which is the same formula as used for calculating the grid momentum from the particle variables. The right-action can be interpreted as a discrete form of particle relabeling, because it corresponds to move the particles in such a way so that the grid velocities remain constant. The chapter also provides some interpretation of these transformations in terms of matrices that determine the local deformation of infinitesimal line elements, which highlights down discrete loop integrals on advected loops. This leads to a discrete circulation theorem.