Jet manifolds provide the conventional language of theory of (nonlinear) differential equations, differential operators and Lagrangian systems on fiber bundles. In the framework of jet formalism, sections of fiber bundles over a manifold X are identified by a finite number of terms of their
series at points of X. A key point is that such equivalence classes of sections constitute a finite-dimensional smooth manifold. This fact enables one to consider a finite-dimensional configuration space of a dynamic system parameterized by a finite number of coordinates, but not ill-defined infinite-dimensional functional spaces. Taylor
Formalism of jet manifolds is about of 50 years old, but it remains unknown for theoreticians. At the same time, classical field theory and non-autonomous classical mechanics are adequately formulated in the terms of jet manifolds.
Moreover, jet manifolds provide the language of modern differential geometry to deal with general connections which are represented by sections of jet bundles. As a consequence, the dynamics of field systems and mechanical systems includes connections in a natural way.
What about describing dynamic systems, theoreticians remain in the middle of last century.
G. Sardanashvily, Five lectures on the jet manifolds in field theory arXiv: hep-th/9411089
G. Sardanashvily, Fibre bundles, jet manifolds and Lagrangian theory. Lectures for theoreticians arXiv: 0908.1886
G. Giachetta, L. Mangiarotti, G. Sardanashvily, Advanced mechanics. Mathematical introduction arXiv: 0911.0411
G. Giachetta, L. Mangiarotti, G. Sardanashvily, Advanced Classical Field Theory (World Scientific, 2009)
G. Giachetta, L. Mangiarotti, G. Sardanashvily, Geometric Formulation of Classical and Quantum Mechanics (World Scientific, 2010)