Abstract
A geometrical formalism for nonlinear nonholonomic Lagrangian systems is developed. The solution of the constrained problem is discussed by using almost product structures along the constraint submanifold. Constrained systems with ideal constraints are also considered, and Chetaev conditions are given in geometrical terms. A Noether theorem is also proved.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bates, L., and Śniatycki, J. (1992). Nonholonomic reduction,Reports on Mathematical Physics,32(1), 99–115.
Bates, L., Graumann, H., and MacDonnel, C. (1996). Examples of gauge conservation laws in nonholonomic systems,Reports on Mathematical Physics,37, 295–308.
Cariñena, J. F., and Rañada, M. F. (1993). Lagrangian systems with constraints: A geometric approach to the method of Lagrange multipliers,Journal of Physics A: Mathematical and General,26, 1335–1351.
Cushman, R., Kemppainen, D., Śniatycki, J., and Bates, L. (1995). Geometry of nonholonomic constraints,Reports on Mathematical Physics,36(2/3), 275–286.
de la Torre Juárez, M. (1996). Cyclic coordinates, symmetries and invariant quantities in systems with non-holonomic constraints, Preprint.
de León, M., and Martín de Diego, D. (1996a). Solving non-holonomic Lagrangian dynamics in terms of almost product structures,Extracta Mathematicae,11(2), 325–347.
de León, M., and Martín de Diego, D. (1996b). Non-holonomic mechanical systems in jet bundles, in Proceedings Third Meeting on Current Ideas in Mechanics and Related Fields, Segovia (Spain), June 19–23, 1995,Extracta Mathematicae,11(1), 127–139.
de León, M., and Martín de Diego, D. (1996c). Almost product structures in mechanics, inDifferential Geometry and Applications, Proceedings Conference, Aug. 28–Sept. 1, 1995, Brno, Czech Republic, Massaryk University, Brno, pp. 539–548.
de León, M., and Martín de Diego, D. (1996d). A symplectic formulation of non-holonomic Lagrangian systems, in Proceedings of the IV Fall Workshop: Differential Geometry and its Applications, Santiago de Compostela, September 17–20, 1995,Anales de Física, Monografías RSEF,3, 125–137.
de León, M., and Martín de Diego, D. (1996e). On the geometry of non-holonomic Lagrangian systems,Journal of Mathematical Physics,37(7), 3389–3414.
de León, M., Marrero, J. C., and Martín de Diego, D. (1997). Non-holonomic Lagrangian systems in jet manifolds,Journal of Physics A: Mathematical and General,30, 1167–1190.
de León, M., and Rodrigues, P. R. (1989).Methods of Differential Geometry in Analytical Mechanics, North-Holland, Amsterdam.
Giachetta, G. (1992). Jet methods in nonholonomic mechanics,Journal of Mathematical Physics,33(5), 1652–1665.
Hertz, H. R. (1894).Gesaammelte Werke. Band III, Der Prinzipien der Mechanik in neuem Zusammenhange dargestellt, Barth, Leipzig [English translation, Dover, New York, 1956].
Koiller, J. (1992). Reduction of some classical non-holonomic systems with symmetry,Archive for Rational Mechanics and Analysis,118, 113–148.
Massa, E., and Pagani, E. (1991). Classical dynamics of non-holonomic systems: A geometric approach,Annales de l'Institut Henri Poincaré: Physique Theorique,55, 511–544.
Massa, E., and Pagani, E. (1995). A new look at classical mechanics of constrained systems, preprint.
Neimark, J., and Fufaev, N. (1972).Dynamics of Nonholonomic Systems, American Mathematical Society, Providence, Rhode Island.
Pironneau, Y. (1982). Sur les liaisions non linéaires déplacement virtuels à travail nul, conditions de Chetaev, inProceedings, “Modern Developments in Analytical Mechanics,” Vol. II, Turin, pp. 671–686.
Rosenberg, R. M. (1977).Analytical Dynamics of Discrete Systems, Plenum Press, New York.
Rumiantsev, V. V. (1978). On Hamilton's principle for nonholonomic systems,Prikladnaya Matematika i Mekhanika,42(3), 387–399 [translation,Journal of Applied Mathematics and Mechanics,42(3), 407–419 (1979)].
Sarlet, W. (1996a). A direct geometrical construction of the dynamics of non-holonomic Lagrangian systems, in Proceedings Third Meeting on Current Ideas in Mechanics and Related Fields, Segovia (Spain), June 19–23, 1995,Extracta Mathematicae,11(1), 202–212.
Sarlet, W. (1996b). The geometry of mixed first and second-order differential equations with applications to non-holonomic mechanics, inDifferential Geometry and Applications, Proceedings Conference, Aug. 28–Sept. 1, 1995, Brno, Czech Republic, Massaryk University, Brno, pp. 641–650.
Sarlet, W., Cantrijn, F., and Saunders, D. J. (1995). A geometrical framework for the study of nonholonomic Lagrangian systems,Journal of Physics A: Mathematical and General,28, 3253–3268.
Saunders, D. J. (1989).The Geometry of Jet Bundles, Cambridge University Press, Cambridge.
Saunders, D. J., Sarlet, W., and Cantrijn, F. (1996). A geometrical framework for the study of non-holonomic Lagrangian systems: II,Journal of Physics A: Mathematical and General,29, 4265–4274.
Valcovici, V. (1958). Une extension des liaisions non holonomes et des principes variationnels,Berichte uber die Verhandlungen der Sachsischen Akademie der Wissenschaften zu Leipzig. Mathematisch-Naturwissenschaftliche Klasse,102, 1–39.
Vershik, A. M., and Faddeev, L. D. (1972). Differential geometry and Lagrangian mechanics with constraints,Soviet Physics-Doklady,17(1), 34–36.
Vershik, A. M., and Gershkovich, V. Ya. (1994). Nonholonomic dynamical systems, geometry of distributions and variational problems, inEncyclopaedia of Mathematical Sciences, Vol. 16,Dynamical Systems, VII, V. I. Arnold and S. P. Novikov, eds., Springer-Verlag, Berlin, pp. 1–81.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
de León, M., Marrero, J.C. & de Diego, D.M. Mechanical systems with nonlinear constraints. Int J Theor Phys 36, 979–995 (1997). https://doi.org/10.1007/BF02435796
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02435796