Classical mechanics and field theory admit comprehensive geometric formulation

Classical non-relativistic mechanics and classical field theory are adequately formulated in geometric terms of fibre bundles Y->X where dim X>1 in field theory and X=R in mechanics.

Configuration space of a classical non-relativistic mechanics is a fibre bundle Y->R over the time axis R. Its velocity space is a first order jet manifold JY of Y. Its phase space is the vertical tangent bundle VY of Y. Connections on a fibre bundle Y->R characterize non-relativistic reference frames.

Geometric formulation of classical field theory is based on a representation of classical fields by sections of fibre bundles Y->X where dim X>1. Their Lagrangians are densities on finite order jet manifolds of Y.  Connections on Y->X also are fields, e.g., gauge fields which are sections of the first order jet bundle JY->Y. In a very general setting, in order to include odd fields, e.g., fermions  and ghosts, field theory is formulated on a graded manifold whose body is a fibre bundle Y->X.

The formulation of relativistic mechanics generalizes that of non-relativistic mechanics. It is phrased in terms of one-dimensional submanifolds of a configuration space. In a case of two-dimensional submanifolds, we come to classical string theory.

References:

G.Giachetta, L.Mangiarotti, G.Sardanashvily, Advanced Classical Field Theory (WS, 2009)
G.Giachetta, L.Mangiarotti, G.Sardanashvily, Geometric Formulation of Classical and Quantum Mechanics (WS, 2010)
G.Sardanashvily, Fibre bundles, jet manifolds and Lagrangian theory. Lectures for theoreticians, arXiv: 0908.1886
G.Sardanashvily, Advanced mechanics. Mathematical introduction arXiv: 0911.0411