  The Ma-Trudinger-Wang curvature --- or cross-curvature --- is an object
arising in the regularity theory of optimal transportation. If the
transportation cost is derived from a Hamiltonian action, we show its
cross-curvature can be expressed in terms of the associated Jacobi fields.
Using this expression, we show the least action corresponding to a harmonic
oscillator has zero cross-curvature, and in particular satisfies the necessary
and sufficient condition \Athreew\ for the continuity of optimal maps. We go on
to study gentle perturbations of the free action by a potential, and deduce
conditions on the potential which guarantee either that the corresponding cost
satisfies the more restrictive condition \Athrees\ of Ma, Trudinger and Wang,
or in some cases has positive cross-curvature. In particular, the quartic
potential of the anharmonic oscillator satisfies \Athrees\ in the perturbative
regime.
