Non-standard connections in classical mechanics

TL;DR

This paper investigates the role of connections in the jet-bundle formulation of classical mechanics, revealing their hidden dependence and conditions under which this dependence affects the dynamics.

Contribution

It demonstrates that connection dependence in time-dependent classical mechanics is generally present but dynamically irrelevant, except for energy variation, and explores the link between first integrals and connections.

Findings

Connection dependence is present in time-dependent classical mechanics.

This dependence is usually hidden by natural trivial connections.

The dependence is dynamically irrelevant except for energy variation.

Abstract

In the jet-bundle description of first-order classical field theories there are some elements, such as the lagrangian energy and the construction of the hamiltonian formalism, which require the prior choice of a connection. Bearing these facts in mind, we analyze the situation in the jet-bundle description of time-dependent classical mechanics. So we prove that this connection-dependence also occurs in this case, although it is usually hidden by the use of the ``natural'' connection given by the trivial bundle structure of the phase spaces in consideration. However, we also prove that this dependence is dynamically irrelevant, except where the dynamical variation of the energy is concerned. In addition, the relationship between first integrals and connections is shown for a large enough class of lagrangians.