In the published version of the paper:

Peter W. Michor, David Mumford:
An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach.
Applied and Computational Harmonic Analysis 23 (2007), 74113. doi:10.1016/j.acha.2006.07.004.
the following error has occurred:
The horizontal geodesic equation in 3.6 should have an overall  in front of
it:
a_t =  \frac.....
The version on Peter Michor's homepage has been corrected.