A complete lift for semisprays
  In this paper, we define a complete lift for semisprays. If $S$ is a semispray on a manifold $M$, its complete lift is a new semispray $S^c$ on $TM$. The motivation for this lift is two-fold: First, geodesics for $S^c$ correspond to the Jacobi fields for $S$, and second, this complete lift generalizes and unifies previously known complete lifts for Riemannian metrics, affine connections, and regular Lagrangians. When $S$ is a spray, we prove that the projective geometry of $S^c$ uniquely determines $S$. We also study how symmetries and constants of motions for $S$ lift into symmetries and constants of motions for $S^c$. 
Ioan Bucataru, Matias F. Dahl