The technique of symplectic manifolds is well known to provide the adequate Hamiltonian formulation of autonomous mechanics. Its realistic example is a mechanical system whose configuration space is a manifold M and whose phase space is the cotangent bundle T*M of M provided with the canonical symplectic form on T*M. Any autonomous Hamiltonian system locally is of this type.
However, this geometric formulation of autonomous mechanics is not extended to mechanics under time-dependent transformations because the symplectic form
Om fails to be invariant
under these transformations. As a palliative variant, one has developed time-dependent
mechanics on a configuration space Q=RxM where R is the time axis. Its
phase space RxT*M is provided
with the pull-back presymplectic form. However, this presymplectic form also is
broken by time-dependent transformations.
We address non-relativistic mechanics in a case of arbitrary time-dependent transformations. Its configuration space is a fibre bundle Q->R. Its velocity space is the first order jet manifold of sections of Q->R. A phase space is the vertical cotangent bundle V*Q of Q->R.
This formulation of non-relativistic mechanics is similar to that of classical field theory on fibre bundles over a base of dimension >
1 (#). A difference between mechanics and field
theory however lies in the fact that connections on bundles over R
are flat, and they fail to be dynamic variables, but describe reference frames.
Advanced mechanics. Mathematical introduction, arXiv: 0911.0411