Nonrelativistic Geodesic Motion

TL;DR

The paper demonstrates that nonrelativistic second order dynamics can be interpreted as geodesic equations with respect to a nonlinear connection, bridging nonrelativistic and relativistic geodesic frameworks.

Contribution

It introduces a geometric interpretation of nonrelativistic mechanics as geodesic motion, connecting it to relativistic concepts through nonlinear connections.

Findings

Nonrelativistic equations are equivalent to geodesic equations on a tangent bundle.

Comparison between relativistic and nonrelativistic geodesic equations is provided.

Analysis of Jacobi vector fields along nonrelativistic geodesics is included.

Abstract

We show that any second order dynamic equation on a configuration space $X\to R$ of nonrelativistic mechanics can be seen as a geodesic equation with respect to some (nonlinear) connection on the tangent bundle $TX\to X$ of relativistic velocities. We compare relativistic and nonrelativistic geodesic equations, and study the Jacobi vector fields along nonrelativistic geodesics.