On the Geodesic Form of Non-Relativistic Dynamic Equations

TL;DR

This paper demonstrates that non-relativistic second order dynamic equations can be reformulated as geodesic equations using non-linear connections, providing a geometric perspective on classical mechanics.

Contribution

It introduces a geometric framework that expresses second order dynamic equations as geodesic equations on tangent bundles, including analysis of quadratic cases and Jacobi fields.

Findings

Dynamic equations are equivalent to geodesic equations with non-linear connections.

Detailed analysis of quadratic dynamic equations.

Construction and investigation of Jacobi vector fields using geometric methods.

Abstract

It is shown that any second order dynamic equation on a configuration bundle $Q\to R$ of non-relativistic mechanics is equivalent to a geodesic equation with respect to a (non-linear) connection on the tangent bundle $TQ\to Q$. The case of quadratic dynamic equations is analyzed in details. The equation for Jacobi vector fields is constructed and investigated by the geometric methods.