The point of Jacobi fields is to study variations of geodesics through geodesics, but the Jacobi equation $D_t^2 J + R(J,\dot\gamma)\dot\gamma=0$ makes sense for any curve $\gamma$, not just for geodesics.
Are solutions to this equation still interesting in other situations? For instance, does a variation of a curve along a "Jacobi field" yield curves with related geodesic curvatures? Maybe this is related to e.g. curve shortening flow or some other already studied notions.