As it was mentioned in my previous post (Mechanics as particular classical field theory), non-relativistic mechanics as like as classical field theory (Classical field theory is complete…) is adequately formulated in the terms of fiber bundles and jet manifolds of their sections.
The geometric formulation of relativistic mechanics involves a more sophisticated technique of jets of one-dimensional submanifolds. In the framework of this formalism, submanifolds of a manifold Q are identified if they are tangent to each other at points of Q with some order. Jets of sections (Well-known mathematics that theoreticians do not know) are particular jets of submanifolds when Q-> R is a fiber bundle and these submanifolds are its sections. In contrast with jets of sections, jets of submanifolds in relativistic mechanics admit arbitrary transformations of time t’= t(q) including the Lorentz transformations, but not only t’=t+const. in the non-relativistic case.
Note that jets of two-dimensional submanifolds provide a formulation of classical string theory.
G. Giachetta, L. Mangiarotti, G. Sardanashvily, Geometric Formulation of Classical and Quantum Mechanics (World Scientific, 2010)
G. Sardanashvily, Relativistic mechanics in a general setting, Int. J. Geom. Methods Mod. Phys. 7 (2010) 1307-1319