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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 {\displaystyle Q\to \mathbb {R} } over the time axis R {\displaystyle \mathbb {R} } coordinated by {\displaystyle }.
This bundle is trivial, but its different trivializations Q = R × M {\displaystyle 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 Γ {\displaystyle \Gamma } on Q → R {\displaystyle Q\to \mathbb {R} } which takes a form Γ i = 0 {\displaystyle \Gamma ^{i}=0} with respect to this trivialization. The corresponding covariant differential ∂ i {\displaystyle \partial _{i}} determines the relative velocity with respect to a reference frame Γ {\displaystyle \Gamma }.
As a consequence, non-autonomous mechanics can be formulated as a covariant classical field theory on X = R {\displaystyle X=\mathbb {R} }. Accordingly, the velocity phase space of non-autonomous mechanics is the jet manifold J 1 Q {\displaystyle J^{1}Q} of Q → R {\displaystyle Q\to \mathbb {R} } provided with the coordinates {\displaystyle }. Its momentum phase space is the vertical cotangent bundle V Q {\displaystyle VQ} of Q → R {\displaystyle Q\to \mathbb {R} } coordinated by {\displaystyle } 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 d t {\displaystyle p_{i}dq^{i}-Hdt}.
One can associate to any Hamiltonian non-autonomous system an equivalent Hamiltonian autonomous system on the cotangent bundle T Q {\displaystyle TQ} of Q {\displaystyle Q} coordinated by {\displaystyle } and provided with the canonical symplectic form; its Hamiltonian is p − H {\displaystyle p-H}.