Discrete momentum maps for lattice EPDiff

We focus on the spatial discretization produced by the Variational Particle-Mesh (VPM) method for a prototype fluid equation the known as the EPDiff equation, which is short for Euler-Poincaré equation associated with the diffeomorphism group (of R, or of a ddimensional manifold Ω). The EPDiff equation admits measure valued solutions, whose dynamics are determined by the momentum maps for the left and right actions of the diffeomorphisms on embedded subspaces of R. The discrete VPM analogs of those dynamics are studied here. Our main results are: (i) a variational formulation for the VPM method, expressed in terms of a constrained variational principle principle for the Lagrangian particles, whose velocities are restricted to a distribution DVPM which is a finite-dimensional subspace of the Lie algebra of vector fields on Ω; (ii) a corresponding constrained variational principle on the fixed Eulerian grid which gives a discrete version of the Euler-Poincaré equation; and (iii) discrete versions of the momentum maps for the left and right actions of diffeomorphisms on the space of solutions.