  In this note, we propose a variational framework in which the minimization of
the acceleration on the group of diffeomorphisms endowed with a right-invariant
metric is well-posed. It relies on constraining the acceleration to belong to a
Sobolev space of higher-order than the order of the metric in order to gain
compactness. It provides the theoretical guarantee of existence of minimizers
which is compulsory for numerical simulations.
