Abstract
This article constitutes an appendix to the book by P. A. Griffiths, Exterior differential systems and the calculus of variations. Birkhäuser, 1983. It especially focusses on the distinction between holonomic and nonholonomic mechanical and variational problems, and indicates how rich and interesting the phenomena are in the nonholonomic case.
Similar content being viewed by others
References
Aleksandrov, A. D.: ‘Studies on the maximum principle I’, Izv. Vyssh. Uchebn. Zaved., no. 5 (1958), 126–157. (In Russian).
Aleksandrov, A. D.: ‘Studies on the maximum principle II’, Izv. Vyssh. Uchebn. Zaved., no. 3 (1959), 3–12. (In Russian).
Aleksandrov, A. D.: ‘Studies on the maximum principle III’, Izv. Vyssh. Uchebn. Zaved., no. 5 (1959), 16–22. (In Russian).
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).
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).
Berezin, F. A.: ‘Hamiltonian formalism in the general Lagrange problem’, Uspekhi Mat. Nauk 29, no. 3 (177) (1974), 183–184. (In Russian).
Birkhoff, G.: Lattice theory, AMS, 1967.
Bröcker, T. and Lander, H.: Differentiable germs and cotastrophes, Cambridge Univ. Press, 1975.
Bourbaki, N.: Lie groups and Lie algebras, Addison-Wesley, 1975. (Translated from the French).
Vagner, V. V.: ‘Geometric interpretation of the motion of nonholonomic dynamical systems’, Trudy Sem. Vektor. i Tenzor. Anal. 5 (1941), 301–327. (In Russian).
Varchenko, A. N.: ‘Obstructions to local equivalence of distributions’, Math. Notes 29 (1981), 479–484. (Original: Mat. Zametki 29 (1981), 939–947).
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).
Vershik, A. M. and Faddeev, L. D.: ‘Lagrange mechanics in an invariant setting’, Probl. Teoret. Fiz. (1975), 129–141. (In Russian).
Vershik, A. M.: ‘Classical and nonclassical dynamics with constraints’, in New in global analysis, Voronezh. Gos. Univ., 1984, pp. 23–48. (In Russian).
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).
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).
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).
Davydov, A. A.: ‘Quasi-Hölderness of the boundary of attainability’. Trudy Sem. Vektor. i Tenzor. Anal. 22 (1985), 25–30. (In Russian).
Dirac, P. A. M.: The principles of quantum mechanics, Clarendon Press, 1947.
Kozlov, V. V.: ‘Dynamics of systems with nonintegrable constraints I’, Vestnik Moskov. Gos. Univ., no. 3 (1982), 92–100. (In Russian).
Kozlov, V. V.: ‘Dynamics of systems with nonintegrable constraints II’, Vestnik Moskov. Gos. Univ., no. 4 (1982), 70–76. (In Russian).
Kozlov, V. V.: ‘Dynamics of systems with nonintegrable constraints III’, Vestnik Moskov. Gos. Univ., no. 3 (1983), 102–111. (In Russian).
Lang, S.: Algebra, Addison-Wesley, 1965.
Lang, S.: Introduction to differentiable manifolds, Interscience, 1962.
Milnor, J.: Morse theory. Princeton Univ. Press, 1963.
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).
Rashevskii, P. K., Geometric theory of partial differential equations, Gostekhizdat, Moskva-Leningrad, 1947. (In Russian).
Rumyantsev, V. V.: ‘On Hamilton's principle for nonholonomic systems’, PMM 42, no. 3 (1978), 387–399. (In Russian).
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).
Suslov, G. K., Theoretical mechanics. Gostekhizdat, Moskva-Leningrad, 1946. (In Russian).
Treves, F.: Introduction to pseudodifferential and Fourier integral operators, 1–2, Plenum, 1980.
Filippov, A. F.: ‘On certain questions in optimal control theory’, Vestnik Moskov. Gos. Univ. 2 (1959), 25–32. (In Russian).
Helgason, S.: Differential geometry and symmetric spaces. Acad. Press. 1962.
Brockett, R. W., Millman, R. S. and Sussman, H. J.: Differential geometric control theory, Birkhäuser, 1983.
Gardner, R. ‘Invariants of Pfaffians systems’, Trans. AMS 126 (1967). 514–533.
Nagano, T.: ‘Linear differential systems with singularities and an application to transitive Lie algebras’. J. Math. Soc. Japan 18, no. 4 (1966), 398–404.
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).
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).
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).
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).
Griffiths, P. A.: Exterior differential systems and the calculus of variations, Birkhäuser, 1983.
Cartan, E.: Lecons sur les invariants intégraux, Hermann, 1924.
Cartan, E.: Les systemes différentielles extérieurs et leurs application géométriques, Hermann, 1945.
Chow., W. L.: ‘Ueber Systeme von linearen partiellen Differentialglcichungen erster Ordnung’, Math. Ann 117, no. 1 (1940), 98–105.
Hermann, R.: ‘E. Cartan's geometric theory of partial differential equations’, Adv. Math. 1 (1965), 265–317.
Sternberg, S.: Lectures on differential geometry, Prentice-Hall, 1964.
Author information
Authors and Affiliations
Additional information
Appendix to the Russian translation of [G]. The letter G followed by a number will refer to the corresponding reference in [G]; the quoted references, as well as [G] itself, can be found at the end of the list of references below.
Rights and permissions
About this article
Cite this article
Vershik, A.M., Gershkovich, V.Y. Nonholonomic problems and the theory of distributions. Acta Appl Math 12, 181–209 (1988). https://doi.org/10.1007/BF00047498
Issue Date:
DOI: https://doi.org/10.1007/BF00047498
AMS subject classifications (1980)
Key words
- Exterior differential system
- nonholonomic variational problem
- nonholonomic mechanical problem
- distribution (subbundle of tangent bundle)
- calculus of variations
- constrained problem
- onconstrained problem
- growth vector
- Lagrange's method
- geometric theory of partial differential equations
- Frobenjus theorem
- optimal control theory
- Lie algebras of vector fields