Several examples of nonholonomic mechanical systems

Swaczyna, Martin

Communications in Mathematics, Tome 19 (2011) no. 1, pp. 27-56

Abstract (VO)

A unified geometric approach to nonholonomic constrained mechanical systems is applied to several concrete problems from the classical mechanics of particles and rigid bodies. In every of these examples the given constraint conditions are analysed, a corresponding constraint submanifold in the phase space is considered, the corresponding constrained mechanical system is modelled on the constraint submanifold, the reduced equations of motion of this system (i.e. equations of motion defined on the constraint submanifold) are presented. Finally, solvability of these equations is discussed and general solutions in explicit form are found.

Détail

Classification : 37J60, 70F25, 70G45, 70G75, 70H30
Keywords: Lagrangian system; constraints; nonholonomic constraints; constraint submanifold; canonical distribution; nonholonomic constraint structure; nonholonomic constrained system; reduced equations of motion (without Lagrange multipliers); Chetaev equations of motion (with Lagrange multipliers)

Bibliographie

[1] Bloch, A.M.: Nonholonomic Mechanics and Control. Springer Verlag, New York 2003 | MR | Zbl

[2] Brdička, M., Hladík, A.: Theoretical Mechanics. Academia, Praha 1987 (in Czech) | MR

[3] Bullo, F., Lewis, A.D.: Geometric Control of Mechanical Systems. Springer Verlag, New York, Heidelberg, Berlin 2004 | MR

[4] Cardin, F., Favreti, M.: On nonholonomic and vakonomic dynamics of mechanical systems with nonintegrable constraints. J. Geom. Phys. 18 1996 295–325 | DOI | MR

[5] Cariñena, J.F., Rañada, M.F.: Lagrangian systems with constraints: a geometric approach to the method of Lagrange multipliers. J. Phys. A: Math. Gen. 26 1993 1335–1351 | DOI | MR

[6] Cortés, J.: Geometric, Control and Numerical Aspects of Nonholonomic Systems. Lecture Notes in Mathematics 1793, Springer, Berlin 2002 | MR | Zbl

[7] Cortés, J., León, M. de, Marrero, J.C., Martínez, E.: Nonholonomic Lagrangian systems on Lie algebroids. Discrete Contin. Dyn. Syst. A 24 2009 213–271 | DOI | MR | Zbl

[8] León, M. de, Marrero, J.C., Diego, D.M. de: Non-holonomic Lagrangian systems in jet manifolds. J. Phys. A: Math. Gen. 30 1997 1167–1190 | DOI | MR

[9] León, M. de, Marrero, J.C., Diego, D.M. de: Mechanical systems with nonlinear constraints. Int. Journ. Theor. Phys. 36, No.4 1997 979–995 | DOI | MR

[10] Giachetta, G.: Jet methods in nonholonomic mechanics. J. Math. Phys. 33 1992 1652–1655 | DOI | MR | Zbl

[11] Janová, J.: A Geometric theory of mechanical systems with nonholonomic constraints. Thesis, Faculty of Science, Masaryk University, Brno, 2002 (in Czech)

[12] Janová, J., Musilová, J.: Non-holonomic mechanics mechanics: A geometrical treatment of general coupled rolling motion. Int. J. Non-Linear Mechanics 44 2009 98–105 | DOI

[13] Koon, W.S., Marsden, J.E.: The Hamiltonian and Lagrangian approaches to the dynamics of nonholonomic system. Reports on Mat. Phys. 40 1997 21–62 | DOI | MR

[14] Krupková, O.: Mechanical systems with nonholonomic constraints. J. Math. Phys. 38 1997 5098–5126 | DOI | MR

[15] Krupková, O.: On the geometry of non-holonomic mechanical systems. O. Kowalski, I. Kolář, D. Krupka, J. Slovák (eds.)Proc. Conf. Diff. Geom. Appl., Brno, August 1998 Masaryk University, Brno 1999 533-546 | MR

[16] Krupková, O.: Recent results in the geometry of constrained systems. Rep. Math. Phys. 49 2002 269–278 | DOI | MR | Zbl

[17] Krupková, O.: The nonholonomic variational principle. J. Phys. A: Math. Theor. 42 2009 No. 185201 | DOI | MR | Zbl

[18] Krupková, O.: Geometric mechanics on nonholonomic submanifolds. Communications in Mathematics 18 2010 51–77 | MR | Zbl

[19] Krupková, O., Musilová, J.: The relativistic particle as a mechanical system with nonlinear constraints. J. Phys. A: Math. Gen. 34 2001 3859–3875 | DOI

[20] Marsden, J.E., Ratiu, T.S.: Introduction to Mechanics and Symmetry. Texts in Applied Mathematics 17, Springer Verlag, New York 1999 2nd ed. | MR | Zbl

[21] Massa, E., Pagani, E.: A new look at classical mechanics of constrained systems. Ann. Inst. Henri Poincaré 66 1997 1–36 | MR | Zbl

[22] Neimark, Ju.I., Fufaev, N.A.: Dynamics of Nonholonomic Systems. Translations of Mathematical Monographs 33, American Mathematical Society, Rhode Island 1972 | Zbl

[23] Sarlet, W., Cantrijn, F., Saunders, D.J.: A geometrical framework for the study of non-holonomic Lagrangian systems. J. Phys. A: Math. Gen. 28 1995 3253–3268 | DOI | MR | Zbl

[24] Swaczyna, M.: On the nonholonomic variational principle. K. Tas, D. Krupka, O.Krupková, D. Baleanu (eds.)Proc. of the International Workshop on Global Analysis, Ankara, 2004 AIP Conference Proceedings, Vol. 729, Melville, New York 2004 297–306 | MR | Zbl

[25] Swaczyna, M.: Variational aspects of nonholonomic mechanical systems. Ph.D. Thesis, Faculty of Science, Palacky University, Olomouc, 2005

[26] Tichá, M.: Mechanical systems with nonholonomic constraints. Thesis, Faculty of Science, University of Ostrava, Ostrava, 2004 (in Czech)

[27] Volný, P.: Nonholonomic systems. Ph.D. Thesis, Faculty of Science, Palacky University, Olomouc, 2004