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 Q → R over the time axis R coordinated by ( t , q i ) .
This bundle is trivial, but its different trivializations Q = R × M correspond to the choice of different non-relativistic reference frames. Such a reference frame also is represented by a connection Γ on Q → R which takes a form Γ i = 0 with respect to this trivialization. The corresponding covariant differential ( q t i − Γ i ) ∂ i determines the relative velocity with respect to a reference frame Γ .
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 X = R . Accordingly, the velocity phase space of non-autonomous mechanics is the jet manifold J 1 Q of Q → R provided with the coordinates ( t , q i , q t i ) . Its momentum phase space is the vertical cotangent bundle V Q of Q → R coordinated by ( t , q i , p i ) and endowed with the canonical Poisson structure. The dynamics of Hamiltonian non-autonomous mechanics is defined by a Hamiltonian form p i d q i − H ( t , q i , p i ) d t .
One can associate to any Hamiltonian non-autonomous system an equivalent Hamiltonian autonomous system on the cotangent bundle T Q of Q coordinated by ( t , q i , p , p i ) and provided with the canonical symplectic form; its Hamiltonian is p − H .
