Hamiltonian autonomous mechanics is well described in the framework of symplectic and Poisson geometry. This description fails to be extended to time-dependent mechanical systems subject to time-dependent transformations.
Lagrangian and Hamiltonian time-dependent non-relativistic mechanics is adequately formulated in terms of fiber bundles Q->R over the time axis R (see References). This formulation is similar to that of classical field theory phrased in the language of fiber bundles Y->X and jet manifolds JY of their sections (Classical field theory is complete…). In particular, the phase space of time-dependent mechanics is the vertical cotangent bundle VQ of its configuration space Q which is analogous to a polysymplectic phase space of covariant Hamiltonian field theory (What is true Hamiltonian field theory).
However, an essential difference between time-dependent non-relativistic mechanics and classical field theory lies in the fact that, in field theory, connections on Y->X are dynamic variables (e.g., gauge fields), whereas connections on Q->R are always flat and, therefore, they are not dynamic variables, but characterize reference frames in non-relativistic mechanics.
Thus, fiber bundles and jet manifolds of their sections provide the comprehensive mathematical formulation both of classical field theory and non-relativistic time-dependent mechanics.
Let us note that a geometric formulation of relativistic mechanics as like as string theory involves a more sophisticated technique of jets of submanifolds.
WikipediA: Non-autonomous mechanics
L. Mangiarotti, G. Sardanashvily, Gauge Mechanics (World Scientific, 1998)
G. Giachetta, L. Mangiarotti, G. Sardanashvily, Geometric Formulation of Classical and Quantum Mechanics (World Scientific, 2011)
G. Sardanashvily, Advanced mechanics. Mathematical introduction arXiv: 0911.0411