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 (in particular, non-autonomous Hamiltonian mechanics) can be formulated as a covariant classical field theory (in particular covariant Hamiltonian 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 .