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 formula_1 over the time axis formula_2 coordinated by formula_3. 
This bundle is trivial, but its different trivializations formula_4 correspond to the choice of different non-relativistic reference frames. Such a reference frame also is represented by a connection
formula_5 on formula_6 which takes a form formula_7 with respect to this trivialization. The corresponding covariant differential formula_8
determines the relative velocity with respect to a reference frame formula_5.
As a consequence, non-autonomous mechanics (in particular, non-autonomous Hamiltonian mechanics) can be formulated as a covariant classical field theory (in particular covariant Hamiltonian field theory) on formula_10. Accordingly, the velocity phase space of non-autonomous mechanics is the jet manifold formula_11 of formula_1 provided with the coordinates formula_13. Its momentum phase space is the vertical cotangent bundle formula_14 of formula_1 coordinated by formula_16 and endowed with the canonical Poisson structure. The dynamics of Hamiltonian non-autonomous mechanics is defined by a Hamiltonian form formula_17. 
One can associate to any Hamiltonian non-autonomous system an equivalent Hamiltonian autonomous system on the cotangent bundle formula_18 of formula_19 coordinated by formula_20 and provided with the canonical symplectic form; its Hamiltonian is formula_21.