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In some implementations of the Large Deformation Diffeomorphic Metric Mappingformulation for image registration we consider the motion of particles whichlocally translate image data. We then lift the motion of the particles toobtain a motion on the entire image. However, it is certainly possible toconsider particles which do more than translate, and this is what will bedescribed in this paper. As the unreduced Lagrangian associated to EPDiffpossesses $\Diff(M)$ symmetry, it must also exhibit $G \subset \Diff(M)$symmetry, for any Lie subgroup. In this paper we will describe a tower of Liegroups $G^{(0)} \subseteq G^{(1)} \subseteq G^{(2)} \subseteq...$ whichcorrespond to preserving $k$-th order jet-data. The reduced configurationspaces $Q^{(k)} := \Diff(M) / G^{(k)}$ will be finite-dimensional (inparticular, $Q^{(0)}$ is the configuration manifold for $N$ particles in $M$).We will observe that $G^{(k)}$ is a normal subgroup of $G^{(0)}$ and so thequotient $G^{(0)} / G^{(k)}$ is itself a (finite dimensional) Lie group whichacts on $Q^{(k)}$. This makes $Q^{(k)}$ a principle bundle over $Q^{(0)}$ andthe reduced geodesic equations on $Q^{(k)}$ will possess $G^{(0)} /G^{(k)}$-symmetry. Noether's theorem implies the existence of conserved momentafor the reduced system on $T^{\ast}Q^{(k)}$.

Symmetries in LDDMM with higher order momentum distributions