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 over the time axis \mathbb R coordinated by (t,q^i). This bundle is trivial, but its different trivializations correspond to the choice of different non-relativistic reference frames. Such a reference frame also is represented by a connection \Gamma on which takes a form \Gamma^i =0 with respect to this trivialization. The corresponding covariant differential 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 provided with the coordinates. Its momentum phase space is the vertical cotangent bundle VQ of 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. One can associate to any Hamiltonian non-autonomous system an equivalent Hamiltonian autonomous system on the cotangent bundle TQ of Q coordinated by and provided with the canonical symplectic form; its Hamiltonian is p-H.