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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

Citations:

Zbl 0512.49003
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[1] Aleksandrov, A. D.: ?Studies on the maximum principle I?, Izv. Vyssh. Uchebn. Zaved., no. 5 (1958), 126-157. (In Russian).
[2] Aleksandrov, A. D.: ?Studies on the maximum principle II?, Izv. Vyssh. Uchebn. Zaved., no. 3 (1959), 3-12. (In Russian).
[3] Aleksandrov, A. D.: ?Studies on the maximum principle III?, Izv. Vyssh. Uchebn. Zaved., no. 5 (1959), 16-22. (In Russian).
[4] Arnol’d, V. I.: ?Singularities in variational calculus?, J. Soviet Math. 27, no. 3 (1984), 2679-2712. (Original: Itogi Nauk. i Tekhn. Sovrem. Probl. Mat. 22 (1983), 3-55). · Zbl 0554.58010
[5] Arnol’d, V. I., Varchenko, A. N. and Gusein-Zade, S. M.: Singularities of differentiable maps, Birkhäuser, 1985. (Translated from the Russian. Vol. II, III to appear).
[6] Berezin, F. A.: ?Hamiltonian formalism in the general Lagrange problem?, Uspekhi Mat. Nauk 29, no. 3 (177) (1974), 183-184. (In Russian). · Zbl 0307.70020
[7] Birkhoff, G.: Lattice theory, AMS, 1967. · Zbl 0153.02501
[8] Bröcker, T. and Lander, H.: Differentiable germs and cotastrophes, Cambridge Univ. Press, 1975.
[9] Bourbaki, N.: Lie groups and Lie algebras, Addison-Wesley, 1975. (Translated from the French). · Zbl 0319.17002
[10] Vagner, V. V.: ?Geometric interpretation of the motion of nonholonomic dynamical systems?, Trudy Sem. Vektor. i Tenzor. Anal. 5 (1941), 301-327. (In Russian). · Zbl 0063.07917
[11] Varchenko, A. N.: ?Obstructions to local equivalence of distributions?, Math. Notes 29 (1981), 479-484. (Original: Mat. Zametki 29 (1981), 939-947). · Zbl 0484.58007
[12] Vosilyus, R. V.: ?Contravariant theory of differential extensions in a model of a space with a connection?, Probl. Geom. 14 (1983), 101-175. (In Russian).
[13] Vershik, A. M. and Faddeev, L. D.: ?Lagrange mechanics in an invariant setting?, Probl. Teoret. Fiz. (1975), 129-141. (In Russian). · Zbl 0518.58015
[14] Vershik, A. M.: ?Classical and nonclassical dynamics with constraints?, in New in global analysis, Voronezh. Gos. Univ., 1984, pp. 23-48. (In Russian).
[15] Vinogradov, A. M.: ?Geometry of nonlinear differential equations?, J. Soviet Math. 17 (1981), 1624-1649. (Original: Itogi Nauk. i Tekhn. Probl. Geom. 11 (1980), 89-134). · Zbl 0475.58025
[16] Gershkovich, V. Ya.: ?Two-sided estimates of metrics generated by absolutely nonholonomic distributions on Riemannian manifolds?, Soviet Math. Dokl. 30 (1984), 506-509. (Original: Dokl. Akad. Nauk SSSR 278 (1984), 1040-1044). · Zbl 0591.53033
[17] Gershkovich, V. Ya.: ?Variational problem with nonholonomic constraint on SO(3)?, in New in global analysis, Voronezh. Gos. Univ., 1984, pp. 149-152. (In Russian).
[18] Davydov, A. A.: ?Quasi-Hölderness of the boundary of attainability?. Trudy Sem. Vektor. i Tenzor. Anal. 22 (1985), 25-30. (In Russian). · Zbl 0593.49028
[19] Dirac, P. A. M.: The principles of quantum mechanics, Clarendon Press, 1947. · Zbl 0030.04801
[20] Kozlov, V. V.: ?Dynamics of systems with nonintegrable constraints I?, Vestnik Moskov. Gos. Univ., no. 3 (1982), 92-100. (In Russian). · Zbl 0501.70016
[21] Kozlov, V. V.: ?Dynamics of systems with nonintegrable constraints II?, Vestnik Moskov. Gos. Univ., no. 4 (1982), 70-76. (In Russian).
[22] Kozlov, V. V.: ?Dynamics of systems with nonintegrable constraints III?, Vestnik Moskov. Gos. Univ., no. 3 (1983), 102-111. (In Russian). · Zbl 0516.70017
[23] Lang, S.: Algebra, Addison-Wesley, 1965.
[24] Lang, S.: Introduction to differentiable manifolds, Interscience, 1962. · Zbl 0103.15101
[25] Milnor, J.: Morse theory. Princeton Univ. Press, 1963. · Zbl 0108.10401
[26] Rashevskii, P. K.: ?On linking two arbitrary points of a completely nonholonomic space by an admissible curve?, Uchen. Zap. Moskov. Ped. Inst. Libknekht, Ser. Fit.-Mat. Nauk, no. 2 (1983), 83-94. (In Russian).
[27] Rashevskii, P. K., Geometric theory of partial differential equations, Gostekhizdat, Moskva-Leningrad, 1947. (In Russian).
[28] Rumyantsev, V. V.: ?On Hamilton’s principle for nonholonomic systems?, PMM 42, no. 3 (1978), 387-399. (In Russian).
[29] Rumyantsev, V. V.: ?On Hamilton’s principle and a generalized method of Hamilton-Jacobi for nonholonomic systems?, Teor. i Primen. Mehanika. no. 4 (1978), 131-138. (In Russian).
[30] Suslov, G. K., Theoretical mechanics. Gostekhizdat, Moskva-Leningrad, 1946. (In Russian).
[31] Treves, F.: Introduction to pseudodifferential and Fourier integral operators, 1-2, Plenum, 1980.
[32] Filippov, A. F.: ?On certain questions in optimal control theory?, Vestnik Moskov. Gos. Univ. 2 (1959), 25-32. (In Russian).
[33] Helgason, S.: Differential geometry and symmetric spaces. Acad. Press. 1962. · Zbl 0111.18101
[34] Brockett, R. W., Millman, R. S. and Sussman, H. J.: Differential geometric control theory, Birkhäuser, 1983.
[35] Gardner, R. ?Invariants of Pfaffians systems?, Trans. AMS 126 (1967). 514-533. · Zbl 0161.41301
[36] Nagano, T.: ?Linear differential systems with singularities and an application to transitive Lie algebras?. J. Math. Soc. Japan 18, no. 4 (1966), 398-404. · Zbl 0147.23502
[37] Evtushik, L. E., Lumiste, U. G., Ostianu, N. M. and Shirokov, A. P.: ?Differential-geometric structures on manifolds?, J. Soviet Math. 14 (1980). 1573-1719. (Original: Itogi Nauk. i Tekhn. Probl. Geom. 9 (1979), 5-246). · Zbl 0455.58002
[38] Arnol’d, V. I. Kozlov, V. V. and Neishtadt, A. I.: ?Mathematical aspects of classical and celestial mechanics?, Itogi Nauk. i Tekhn. Sovrem. Probl. Mat. Fund. Napravl. Dinam. Sistem. 3 (1985). (In Russian).
[39] Vershik, A. M. and Gershkovich, V. Ya.. ?Nonholonomic dynamical systems. Geometry of distributions and variational problems?, Itogi Nauk. i Tekhn. Sovrem. Probl. Mat. Fund. Napravl. Dinam. Sistem. 7 (1986), 5-85. (In Russian). · Zbl 0797.58007
[40] Vershik, A. M. and Gershkovich, V. Ya.: ?Nonholonomic dynamical systems. Geometry of distributions and variational problems?, Itogi Nauk. i Tekhn. Sovrem. Probl. Mat. Fund. Napravl. Dinam. Sistem 8 (1986). (In Russian). · Zbl 0797.58007
[41] Griffiths, P. A.: Exterior differential systems and the calculus of variations, Birkhäuser, 1983. · Zbl 0512.49003
[42] Cartan, E.: Lecons sur les invariants intégraux, Hermann, 1924. · JFM 50.0479.01
[43] Cartan, E.: Les systemes différentielles extérieurs et leurs application géométriques, Hermann, 1945.
[44] Chow., W. L.: ?Ueber Systeme von linearen partiellen Differentialglcichungen erster Ordnung?, Math. Ann 117, no. 1 (1940), 98-105. · Zbl 0022.02304
[45] Hermann, R.: ?E. Cartan’s geometric theory of partial differential equations?, Adv. Math. 1 (1965), 265-317. · Zbl 0142.07104
[46] Sternberg, S.: Lectures on differential geometry, Prentice-Hall, 1964. · Zbl 0129.13102
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