Deviation differential equations. Jacobi fields

TL;DR

This paper introduces deviation differential equations, called Jacobi fields, as vertical extensions of original equations on fibre bundles, providing a unified framework for analyzing their solutions in a general setting.

Contribution

It generalizes the concept of Jacobi fields to a broad class of differential equations on fibre bundles, including Euler-Lagrange and Hamilton equations, via vertical extensions.

Findings

Jacobi fields are solutions to deviation equations on vertical tangent bundles.

Deviation of Euler-Lagrange equations corresponds to Euler-Lagrange equations of the extended Lagrangian.

Covariant Hamilton equations extend to vertical extensions of Hamiltonian forms.

Abstract

Given a differential equation on a smooth fibre bundle Y, we consider its canonical vertical extension to that, called the deviation equation, on the vertical tangent bundle VY of Y. Its solutions are Jacobi fields treated in a very general setting. In particular, the deviation of Euler--Lagrange equations of a Lagrangian L on a fibre bundle Y are the Euler-Lagrange equations of the canonical vertical extension of L onto VY. Similarly, covariant Hamilton equations of a Hamiltonian form H are the Hamilton equations of the vertical extension VH of H onto VY.