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 ${\displaystyle \Gamma }$.

...
Wikipedia