Non-autonomous mechanics

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 ${\displaystyle \mathrm{\Gamma }}$ on ${\displaystyle Q\to \mathbb{R}}$ which takes a form ${\displaystyle {\mathrm{\Gamma }}^i=0}$ with respect to this trivialization. The corresponding covariant differential ${\displaystyle (q_t^i-{\mathrm{\Gamma }}^i){\mathrm{\partial }}_i}$ determines the relative velocity with respect to a reference frame ${\displaystyle \mathrm{\Gamma }}$.

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 ${\displaystyle p_{i}d{q}^i-H(t,q^i,p_i)dt}$.

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 ${\displaystyle p-H}$.