First order deviation equations in spaces with a transport along paths

TL;DR

This paper derives coordinate-free deviation equations for two particles in a manifold with various geometric structures, providing insights into their relative motion and energy in a generalized setting.

Contribution

It introduces a unified, coordinate-free framework for deviation equations considering linear transport, connection, and metric structures on the tangent bundle.

Findings

Derived deviation equations for velocity, momentum, acceleration, and energy

Established approximate relations between these quantities

Extended classical deviation concepts to manifolds with complex geometric structures

Abstract

In a coordinate free form are found the (deviation) equations satisfied by the (infinitesimal) deviation vector, relative velocity, relative momentum, relative acceleration and relative energy of two point particles in a differentiable manifold the tangent bundle of which is endowed with a linear transport along paths, a linear connection and, in the last case, also with a metric. Some approximate relations between these quantities are obtained.