Jacobi Fields for a Nonholonomic Distribution

The variational problem with nonholonomic constraints was considered in detail by Bliss. A distribution is a special case of constraints. Horizontal geodesics on a manifold with flat metric and constant tensor of nonholonomity are considered. It is proved that, in the classical adjoint problem, conjugate points appear, which does not involve any loss of optimality. The second variation of the length (or energy) functional of admissible (horizontal) geodesics for a distribution on a smooth manifold is expressed in terms of the distribution curvature tensor.