Dynamic Connections in Analytical Mechanics

TL;DR

This paper explores the geometric structure of non-relativistic mechanics, showing how dynamic equations relate to connections on bundles, and examines the link between relativistic and non-relativistic equations of motion.

Contribution

It introduces a geometric framework connecting dynamic equations to affine and tangent bundle connections, unifying non-relativistic mechanics with geometric concepts.

Findings

Dynamic equations correspond to geodesic equations on tangent bundles.

Connections on affine jet bundles encode non-relativistic dynamics.

Relates relativistic and non-relativistic equations via geometric structures.

Abstract

It is shown that any dynamic equation on a configuration bundle $Q\to R$ of non-relativistic time-dependent mechanics is associated with connections on the affine jet bundle $J^1Q\to Q$ and on the tangent bundle $TQ\to Q$. As a consequence, any non-relativistic dynamic equation can be seen as a geodesic equation with respect to a (non-linear) connection on the tangent bundle $TQ\to Q$. Using this fact, the relationship between relativistic and non-relativistic equations of motion is studied. The geometric notions of reference frames and relative accelerations in non-relativistic mechanics are introduced in the terms of connections. The covariant form of non-relativistic dynamic equations is written.