Non-autonomous mechanics describe non-relativistic mechanical systems subject to time-dependent transformations. In particular, this is the case of mechanical systems whose Lagrangians and Hamiltonians depend on the time. The configuration space of non-autonomous mechanics is a fiber bundle over the time axis coordinated by .
This bundle is trivial, but its different trivializations correspond to the choice of different non-relativistic reference frames. Such a reference frame also is represented by a connection on which takes a form with respect to this trivialization. The corresponding covariant differential determines the relative velocity with respect to a reference frame .