  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-Poincar\'e equation associated with
the diffeomorphism group (of $\mathbb{R}^d$, or of a $d$-dimensional manifold
$\Omega$). 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 $\mathbb{R}^d$. 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 $D_{\VPM}$ which is a finite-dimensional
subspace of the Lie algebra of vector fields on $\Omega$; (ii) a corresponding
constrained variational principle on the fixed Eulerian grid which gives a
discrete version of the Euler-Poincar\'e equation; and (iii) discrete versions
of the momentum maps for the left and right actions of diffeomorphisms on the
space of solutions.
