Non-autonomous mechanics
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.
As a consequence, non-autonomous mechanics can be formulated as a covariant classical field theory on. Accordingly, the velocity phase space of non-autonomous mechanics is the jet manifold of provided with the coordinates. Its momentum phase space is the vertical cotangent bundle of coordinated by and endowed with the canonical Poisson structure. The dynamics of Hamiltonian non-autonomous mechanics is defined by a Hamiltonian form.
One can associate to any Hamiltonian non-autonomous system an equivalent Hamiltonian autonomous system on the cotangent bundle of coordinated by and provided with the canonical symplectic form; its Hamiltonian is.
