A variational framework for higher order perturbations

A covariant, global, variational framework for perturbations in field theories is presented. Perturbations are obtained as vertical vector fields on the configuration bundle and they drag, exactly, solution into solutions. The flow of a perturbation drags solutions into solutions and the dragged perturbed solutions can be expanded in a series with respect to the flow parameter, hence it contains perturbations at any order. Mechanics is included as a special case. As a simple application, we recover the well-known discussion about stability of geodesics on a sphere $S_2$.