K-Jet field approximations to geodesic deviation equations

Abstract

Let [Formula: see text] be a smooth manifold and [Formula: see text] a semi-spray defined on a sub-bundle [Formula: see text] of the tangent bundle [Formula: see text]. In this work, it is proved that the only non-trivial [Formula: see text]-jet approximation to the exact geodesic deviation equation of [Formula: see text], linear on the deviation functions and invariant under an specific class of local coordinate transformations, is the Jacobi equation. However, if the linearity property on the dependence in the deviation functions is not imposed, then there are differential equations whose solutions admit [Formula: see text]-jet approximations and are invariant under arbitrary coordinate transformations. As an example of higher-order geodesic deviation equations, we study the first- and second-order geodesic deviation equations for a Finsler spray.