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 $$Q\to \mathbb R$$ over the time axis $$\mathbb R$$ coordinated by $$(t,q^i)$$.

This bundle is trivial, but its different trivializations $$Q=\mathbb R\times M$$ correspond to the choice of different non-relativistic reference frames. Such a reference frame also is represented by a connection $$\Gamma$$ on $$Q\to\mathbb R$$ which takes a form $$\Gamma^i =0$$ with respect to this trivialization. The corresponding covariant differential $$(q^i_t-\Gamma^i)\partial_i$$ determines the relative velocity with respect to a reference frame $$\Gamma$$.

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 $$X=\mathbb R$$. Accordingly, the velocity phase space of non-autonomous mechanics is the jet manifold $$J^1Q$$ of $$Q\to \mathbb R$$ provided with the coordinates $$(t,q^i,q^i_t)$$. Its momentum phase space is the vertical cotangent bundle $$VQ$$ of $$Q\to \mathbb R$$ 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 $$p_idq^i-H(t,q^i,p_i)dt$$.

One can associate to any Hamiltonian non-autonomous system an equivalent Hamiltonian autonomous system on the cotangent bundle $$TQ$$ of $$Q$$ coordinated by $$(t,q^i,p,p_i)$$ and provided with the canonical symplectic form; its Hamiltonian is $$p-H$$.