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

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

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 ${\displaystyle X=\mathbb {R} }$. Accordingly, the velocity phase space of non-autonomous mechanics is the jet manifold ${\displaystyle J^{1}Q}$ of ${\displaystyle Q\to \mathbb {R} }$ provided with the coordinates ${\displaystyle (t,q^{i},q_{t}^{i})}$. Its momentum phase space is the vertical cotangent bundle ${\displaystyle VQ}$ of ${\displaystyle Q\to \mathbb {R} }$ coordinated by ${\displaystyle (t,q^{i},p_{i})}$ and endowed with the canonical Poisson structure. The dynamics of Hamiltonian non-autonomous mechanics is defined by a Hamiltonian form ${\displaystyle p_{i}dq^{i}-H(t,q^{i},p_{i})dt}$.

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