  We consider the dynamics of vector fields on three-manifolds which are
constrained to lie within a plane field, such as occurs in nonholonomic
dynamics. On compact manifolds, such vector fields force dynamics beyond that
of a gradient flow, except in cases where the underlying manifold is
topologically simple. Furthermore, there are strong restrictions on the types
of gradient flows realized within plane fields: such flows lie on the boundary
of the space of nonsingular Morse-Smale flows. This relationship translates to
knot-theoretic obstructions for the link of singularities in the flow. In the
case of an integrable plane field, the restrictions are even finer, forcing
taut foliations on surface bundles. The situation is completely different in
the case of contact plane fields, however: it is easy to realize gradient
fields within overtwisted contact structures (the nonintegrable analogue of a
foliation with Reeb components).
