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.
References:
WikipediA: Non-autonomous_mechanics
Advanced mechanics. Mathematical introduction, arXiv: 0911.0411
