  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 $S^c$.
