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

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

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

provided with the coordinates

Its momentum phase space is the vertical cotangent bundle

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

{\displaystyle TQ}

and provided with the canonical symplectic form; its Hamiltonian is

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