Hamiltonian Geodesics in Nonholonomic Differential Systems

Let a smooth manifold M be equipped with a smoothly varying positive definite quadratic form on a distribution-subbundle V of the tangent bundle TM. One assumes that V is completely nonintegrable, equivalently all sections of V together with all brackets between them generate TM at each point of M. Then by the well-known theorem of Rashevski-Chow [3] it is possible to connect any two points (in a connected component of M) by a piecewise smooth curve which is tangent to V playing the role of nonholonomic constraint [15]. Having any Riemannian metric on M, we can endow M with a metric dv defined to be the infimum of the lengths of all such curves joining two points. This metric is called the Carnot-Caratheodory metric [9] or sub-Riemannian metric. It is expressed by a contravariant metric tensor gij(x) : T’M + TM, which is nonnegative definite and its image defines the distribution V. Variational problems with nonholonomic constraints offer questions of great mathematical and physical interest [15, 21. One of such questions concerns the local structure of Exp-map and the generic properties of systems of geodesics rays forming the corresponding wavefronts. The problem of the shortest paths in sub-Riemannian manifold with boundary provides new singular Lagrangian varieties of geodesics analogously to the classification of systems of gliding rays in the Riemannian case (cf. ]71).