A Special Nonlinear Connection in Second Order Geometry

TL;DR

This paper introduces a special nonlinear connection in second order geometry, derived from deviation equations of mechanical systems with external forces, with applications to Jacobi equations in Finsler and Riemann spaces.

Contribution

It establishes a new nonlinear connection in second order geometry based on deviation equations, linking mechanical systems and geometric structures.

Findings

Derivation of a nonlinear connection from deviation equations.

Application to Jacobi equations in Finsler and Riemann spaces.

Insight into geometric structures of second order tangent bundles.

Abstract

We show that, for mechanical system with external forces, the equations of deviations of solution curves of the corresponding Lagrange equations,determine a nonlinear connection on the second order osculator (second order tangent) bundle. In particular, Jacobi equations in Finsler and Riemann spaces determine such a nonlinear connection.