  We use the Hamiltonian formulation of the geodesic equation in the
Schwarzschild space-time so as to get the variational equation as the
counterpart of the Jacobi equation in this approach. In this context we are
able to apply the Morales-Ramis theorem to link the integrability of the
geodesic equation to the integrability, in the sense of differential Galois
theory, of the variational equation. This link is strong enough to hold even on
geodesics for which the usual conserved quantities fail to be independent, as
is the case of circular geodesics. We show explicitly the particular cases of
some null geodesics and their variational equations.
