## Nonholonomic problems and the theory of distributions.(English)Zbl 0666.58004

The Russian translation of the book by P. A. Griffiths on “Exterior differential systems and the calculus of variations” (Birkhäuser, 1983; Zbl 0512.49003) has been supplemented by an appendix on nonholonomic problems. The present paper is an English translation of this appendix.
Most of the paper deals with variational nonholonomic problems (as opposed to mechanical nonholonomic problems), i.e. two-point variational problems for which the tangent vectors to admissible curves are bound to lie in a non-integrable distribution. Section 2 contains generalities on distributions and codistributions. The concept of a completely non- holonomic differential system is introduced, together with its degree of nonholonomy and its associated growth vector. A theorem states that nonholonomic distributions generically have maximal growth vectors. Technical material on quotients of free Lie algebras in the last section makes it possible to enter into a classification of such generic cases.
Section 3 is the main one. The Euler-Lagrange equations corresponding to a nonholonomic variational problem are described. The concept of a nonholomic metric is introduced with the corresponding generalization of the exponential map. Considering an $$\epsilon$$-ball in this metric, i.e. a set of points that can be reached by an admissible curve of length $$\leq \epsilon$$, the geometry of its boundary is studied. It is argued that such an $$\epsilon$$-sphere does not consist of the ends of geodesics of length $$\epsilon$$ (a “wave front”) and has singularities. A full description of the geometry of such spheres is given for the case that the manifold is SO(3) or the Heisenberg group.
Section 4 discusses the entirely different situation of nonholonomic constraints in mechanics, where an invariant formulation of d’Alembert’s principle leads to the concept of geodesics of a truncated connection.
Reviewer: W.Sarlet

### MSC:

 58A30 Vector distributions (subbundles of the tangent bundles) 49Q99 Manifolds and measure-geometric topics 58A15 Exterior differential systems (Cartan theory) 70F25 Nonholonomic systems related to the dynamics of a system of particles

Zbl 0512.49003
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