  The Jacobi equation in pseudo-Riemannian geometry determines the linearized
geodesic flow. The linearization ignores the relative velocity of the
geodesics. The generalized Jacobi equation takes the relative velocity into
account; that is, when the geodesics are neighboring but their relative
velocity is arbitrary the corresponding geodesic deviation equation is the
generalized Jacobi equation. The Hamiltonian structure of this nonlinear
equation is analyzed in this paper. The tidal accelerations for test particles
in the field of a plane gravitational wave and the exterior field of a rotating
mass are investigated. In the latter case, the existence of an attractor of
uniform relative radial motion with speed $2^{-1/2}c\approx 0.7 c$ is pointed
out. The astrophysical implications of this result for the terminal speed of a
relativistic jet is briefly explored.
