Abstract
Two types of nonholonomic systems with symmetry are treated: (i) the configuration space is a total space of a G-principal bundle and the constraints are given by a connection; (ii) the configuration space is G itself and the constraints are given by left-invariant forms. The proofs are based on the method of quasicoordinates. In passing, a derivation of the Maurer-Cartan equations for Lie groups is obtained. Simple examples are given to illustrate the algorithmical character of the main results.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arnold, V. I., Mathematical Methods of Classical Mechanics, Springer, 1978.
Arnold, V. I., Kozlov, V. V., & Neishtadt, A. I., Dynamical Systems III, Springer, 1989.
Abraham, R. & Marsden, J., Foundations of Mechanics, Benjamin, 1978.
Appel, P., Traité de Mécanique Rationnelle, Gauthier-Villars, 1953.
Brockett, R., Control theory and Singular Riemannian Geometry, in New Directions in Applied Mathematics, P. J. Hilton & G. S. Young, eds., Springer, 1982.
Baranov, Yu. S. & Gliklikh, Yu. E., A mechanical connection on the group of volume preserving diffeomorphisms, Func. Anal. Appl. 22, 133–135, 1988.
Beghin, H., Etude théorique des compas gyrostatiques, Anschutz et Sperry, Imprimerie Nationale, Paris, 1921.
Bloch, A. M. & McClamroch, N., Control of mechanical systems with classical nonholonomic constraints, preprint, Ohio State University.
Burns, J. A., Ball rolling on a turntable: analog for charged particle dynamics, Amer. J. Phys. 49, 5658, 1981.
Cartan, E., Sur la représentation géométrique des systémes matériels non holonomes, Proc. Int. Cong. Math., vol. 4, Bologna, 1928, 253–261.
Cantrijn, F., Carinena, J. F., Crampin, M. & Ibort, L. A., Reduction of degenerate Lagrangian systems, J. Geom. Phys. 3, 353–400, 1986.
Choquet-Bruhat, Y., DeWitt Morette, C. & Dillard-Bleick, M., Analysis, Manifolds and Physics, North-Holland, 1987.
Eden, R. J., The quantum mechanics of nonholonomic systems, Proc. Roy. Soc. London A 205, 583–595, 1951.
Godbillon, C., Géométrie Différentielle et Mécanique Analytique, Hermann, 1969.
Gershkovich, V. & Vershik, A., Nonholonomic manifolds and nilpotent analysis, J. Geom. Phys. 5, 407–452, 1988.
Hamel, G., Theoretische Mechanik, Springer, 1949.
Hertz, H. R., The Principles of Mechanics, Dover, 1956.
Hicks, N., Notes on Differential Geometry, Van Nostrand, 1965.
Jankowski, K. & Maryniak, J., Le systéme commandé en tant que systéme a liaisons non holonomes, J. Méc. Th. Appl., 7, 157–173, 1988.
Kane, T., AUTOLEV software, Stanford University.
Koiller, J., Nonholonomic systems with symmetry (and a model for a classical particle on a Yang-Mills field), Proc. XV Int. Colloq. on Group Theoretical Methods in Physics, R. Gilmore, ed., World Scientific, 1987.
Kozlov, V. V., Realization of nonintegrable constraints in classical mechanics, Sov. Phys. Dokl. 28, 735–737, 1983.
Kummer, M., On the construction of the reduced phase space of a Hamiltonian systems with symmetry, Indiana Univ. Math. J. 30, 281–291, 1981.
Lagrange, J. L., Mécanique Analytique, Oeuvres, v. 11, Gauthier-Villars, 1888.
de Leon, M. & Rodrigues, P. R., Methods of Differential Geometry in Analytical Mechanics, North-Holland Math. Studies 158, 1989.
Mitchell, J., A local study of Carnot-Carathéodory metrics, Thesis, State Univ. New York, Stony Brook, 1982.
Montgomery, R., Isoholonomic problems and some applications, Commun. Math. Phys. 128, 565–592, 1990.
Marsden, J. & Weinstein, A., Reduction of symplectic manifolds with symmetry, Rep. Math. Phys. 5, 121–130, 1974.
Neimark, Yu. & Fufaev, N. A., Dynamics of Nonholonomic Systems. Amer. Math. Soc. Translations 33, 1972.
Olver, P. J., Applications of Lie groups to Differential Equations, Springer, 1986.
Olsson, R., Some remarks on a paper by C. Carathéodory, Proc. Int. Cong. Math., Amsterdam, 1954, p. 349.
Pansu, P., Géométrie du groupe d'Heisenberg, Thése, Paris VII, 1982.
Poincaré, H., Les idées de Hertz sur la mécanique, Oeuvres VII, 231–250, Gauthier-Villars, 1952.
Pitanga, P., Projector constrained quantum dynamics, Nuovo Cimenta 101 A, 345–351, 1989 (see also Nuovo Cimenta 103 A, 1529–1533, 1990).
Siegel, C. & Moser, J., Lectures on Celestial Mechanics, Springer, 1971.
Sommerfeld, A., Mechanics, Academic Press, 1952.
Spivak, M., A Comprehensive Introduction to Differential Geometry, vol. 2., Publish or Perish, 1970.
Sattinger, D. H. & Weaver, O. L., Lie groups and algebras with applications to Physics, Geometry and Mechanics, Springer, 1986.
Stanchenko, S. V., Nonholonomic Chaplygin systems, Prikl. Mat. Mekh. 53, 11–17, 1989.
Tatarinov, Ya. V., Consequences of a nonintegrable perturbation of the integrable constraints: model problems of low dimensionality, Prikl. Mat. Mech. 51, 579–586, 1987.
Vershik, A. M., Classical and non-classical dynamics with constraints, in Springer Lect. Notes Math. 1108, ed. Yu. G. Borisovich & Xu. E. Gliklikh, 1984.
Vershik, A. M. & Fadeev, L. D., Lagrangian mechanics in invariant form, Selecta Math. Sov. 1, 339–350, 1981.
Vershik, A. M. & Gershkovich, V. Ya., Geodesic flows on SL(2,R) with nonholonomic restrictions, Sem. Inst. Steklov 155, 7–17, 1986.
Veselov, A. & Veselova, L., Currents on Lie groups with nonholonomic connection and integrable nonhamiltonian systems, Funct. Anal. Appl. 20, 308–309, 1987.
Weinstein, A., The local structure of Poisson manifolds, J. Diff. Geom. 18, 523–557, 1983.
Weber, R. W., Hamiltonian systems with constraints and their meaning in mechanics, Arch. Rational Mech. Anal. 91, 309–335, 1986.
Author information
Authors and Affiliations
Additional information
Communicated by R. McGehee
Rights and permissions
About this article
Cite this article
Koiller, J. Reduction of some classical non-holonomic systems with symmetry. Arch. Rational Mech. Anal. 118, 113–148 (1992). https://doi.org/10.1007/BF00375092
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00375092