Deviation differential equations. Jacobi fields

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.