Article
Full entry |
PDF
(0.4 MB)
Feedback
PDF
(0.4 MB)
Feedback
Summary:
In this survey article, nonholonomic mechanics is presented as a part of geometric mechanics. We follow a geometric setting where the constraint manifold is a submanifold in a jet bundle, and a nonholonomic system is modelled as an exterior differential system on the constraint manifold. The approach admits to apply coordinate independent methods, and is not limited to Lagrangian systems under linear constraints. The new methods apply to general (possibly nonconservative) mechanical systems subject to general (possibly nonlinear) nonholonomic constraints, and admit a straightforward generalization to higher order mechanics and field theory. In particular, we are concerned with the following topics: the geometry of nonholonomic constraints, equations of motion of nonholonomic systems on constraint manifolds and their geometric meaning, a nonholonomic variational principle, symmetries, a nonholonomic Noether theorem, regularity, and nonholonomic Hamilton equations.
References:
[1] Balseiro, P., Marrero, J.C., de Diego, D. Martín, Padrón, E.: A unified framework for mechanics: Hamilton-Jacobi equation and applications. Nonlinearity 23 2010 1887–1918 DOI 10.1088/0951-7715/23/8/006 | MR 2669632
[2] Bloch, A.M.: Nonholonomic Mechanics and Control. Springer Verlag, New York 2003 MR 1978379 | Zbl 1045.70001
[3] Bloch, A.M., Fernandez, O.E., Mestdag, T.: Hamiltonization of nonholonomic systems and the inverse problem of the calculus of variations. Rep. Math. Phys. 63 2009 225–249 DOI 10.1016/S0034-4877(09)90001-5 | MR 2519467 | Zbl 1207.37045
[4] Bloch, A.M., Krishnaprasad, P.S., Marsden, J.E., Murray, R.M.: Nonholonomic systems with symmetry. Arch. Ration. Mech. Anal. 136 1996 21–99 DOI 10.1007/BF02199365 | MR 1423003
[5] Cantrijn, F., de León, M., Marrero, J.C., de Diego, D. Martín: Reduction of constrained systems with symmetry. J. Math. Phys. 40 1999 795–820 DOI 10.1063/1.532686 | MR 1674283
[6] Cardin, F., Favretti, M.: On nonholonomic and vakonomic dynamics of mechanical systems with nonintegrable constraints. J. Geom. Phys. 18 1996 295–325 DOI 10.1016/0393-0440(95)00016-X | MR 1383219 | Zbl 0864.70007
[7] 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 10.1088/0305-4470/26/6/016 | MR 1212006
[8] Cendra, H., Ferraro, S., Grillo, S.: Lagrangian reduction of generalized nonholonomic systems. J. Geom. Phys. 58 2008 1271–1290 DOI 10.1016/j.geomphys.2008.05.002 | MR 2453664 | Zbl 1147.70008
[9] Chetaev, N.G.: On the Gauss principle. Izv. Kazan. Fiz.-Mat. Obsc. 6 1932–33 323–326 (in Russian)
[10] Cortés, J.: Geometric, Control and Numerical Aspects of Nonholonomic Systems. Lecture Notes in Mathematics, Vol. 1793, Springer Verlag, New York 2002 DOI 10.1007/b84020 | MR 1942617 | Zbl 1009.70001
[11] Cortés, J., de León, M., Marrero, J.C., Martínez, E.: Nonholonomic Lagrangian systems on Lie algebroids. Discrete Contin. Dyn. Syst. A 24 2009 213–271 DOI 10.3934/dcds.2009.24.213 | MR 2486576 | Zbl 1161.70336
[12] Cortés, J., de León, M., de Diego, D. Martín, Martínez, S.: Geometric description of vakonomic and nonholonomic dynamics. Comparison of solutions. SIAM J. Control Optim. 41 2003 1389–1412 DOI 10.1137/S036301290036817X | MR 1971955
[13] de León, M., de Diego, D.M.: On the geometry of non-holonomic Lagrangian systems. J. Math. Phys. 37 1996 3389–3414 DOI 10.1063/1.531571 | MR 1401231
[14] de León, M., Marrero, J.C., de Diego, D.M.: Non-holonomic Lagrangian systems in jet manifolds. J. Phys. A: Math. Gen. 30 1997 1167–1190 DOI 10.1088/0305-4470/30/4/018 | MR 1449273
[15] Giachetta, G.: Jet methods in nonholonomic mechanics. J. Math. Phys. 33 1992 1652–1665 DOI 10.1063/1.529693 | MR 1158984 | Zbl 0758.70010
[16] Goldschmidt, H., Sternberg, S.: The Hamilton-Cartan formalism in the calculus of variations. Ann. Inst. Fourier 23 1973 203–267 DOI 10.5802/aif.451 | MR 0341531 | Zbl 0243.49011
[17] Grabowska, K., Grabowski, J.: Variational calculus with constraints on general algebroids. J. Phys. A: Math. Theor. 41 2008 No. 175204 DOI 10.1088/1751-8113/41/17/175204 | MR 2451669 | Zbl 1137.70011
[18] Helmholtz, H.: Ueber die physikalische Bedeutung des Prinzips der kleinsten Wirkung. J. für die reine u. angewandte Math. 100 1887 137–166
[19] Janová, J., Musilová, J.: Non-holonomic mechanics: A geometrical treatment of general coupled rolling motion. Int. J. Non-Linear Mechanics 44 2009 98–105 DOI 10.1016/j.ijnonlinmec.2008.09.002 | Zbl 1203.70036
[20] Koon, W.S., Marsden, J.E.: The Hamiltonian and Lagrangian approaches to the dynamics of nonholonomic systems. Reports on Math. Phys. 40 1997 21–62 DOI 10.1016/S0034-4877(97)85617-0 | MR 1492413 | Zbl 0929.70009
[21] Krupka, D.: Some geometric aspects of variational problems in fibered manifolds. Folia Fac. Sci. Nat. UJEP Brunensis 14 1973 1–65 Electronic transcription: arXiv:math-ph/0110005
[22] Krupka, D.: A geometric theory of ordinary first order variational problems in fibered manifolds, II. Invariance. J. Math. Anal. Appl. 49 1975 469–476 DOI 10.1016/0022-247X(75)90190-0 | MR 0362398 | Zbl 0312.58003
[23] Krupka, D.: Global variational theory in fibred spaces. D. Krupka, D. Saunders (eds.)Handbook of Global Analysis Elsevier 2008 773–836 MR 2389646 | Zbl 1236.58026
[24] Krupka, D., Krupková, O., Prince, G., Sarlet, W.: Contact symmetries of the Helmholtz form. Diff. Geom. Appl. 25 2007 518–542 DOI 10.1016/j.difgeo.2007.06.003 | MR 2351428
[25] Krupka, D., Musilová, J.: Hamilton extremals in higher order mechanics. Arch. Math. (Brno) 20 1984 21–30 MR 0785043
[26] Krupková, O.: Lepagean 2-forms in higher order Hamiltonian mechanics, I. Regularity. Arch. Math. (Brno) 22 1986 97–120 MR 0868124
[27] Krupková, O.: Lepagean 2-forms in higher order Hamiltonian mechanics, II. Inverse Problem. Arch. Math. (Brno) 23 1987 155–170 MR 0930318
[28] Krupková, O.: The Geometry of Ordinary Variational Equations. Lecture Notes in Mathematics, Vol. 1678. Springer Verlag, Berlin 1997 MR 1484970
[29] Krupková, O.: Mechanical systems with nonholonomic constraints. J. Math. Phys. 38 1997 5098–5126 DOI 10.1063/1.532196 | MR 1471916
[30] Krupková, O.: On the geometry of non-holonomic mechanical systems. O. Kowalski, I. Kolář, D. Krupka, J. Slovák (eds.)Differential Geometry and Applications Proc. Conf. Brno 1998. Masaryk Univ., Brno 1999 533–546 MR 1708942
[31] Krupková, O.: Higher-order mechanical systems with constraints. J. Math. Phys. 41 2000 5304–5324 MR 1770957
[32] Krupková, O.: Differential systems in higher-order mechanics. D. Krupka (ed.)Proceedings of the Seminar on Differential Geometry Mathematical Publications, Vol. 2. Silesian Univ., Opava 2000 87–130 MR 1855571
[33] Krupková, O.: Recent results in the geometry of constrained systems. Reports on Math. Phys. 49 2002 269–278 DOI 10.1016/S0034-4877(02)80025-8 | MR 1915806 | Zbl 1018.37041
[34] Krupková, O.: Partial differential equations with differential constraints. J. Differential Equations 220 2006 354–395 DOI 10.1016/j.jde.2005.03.003 | Zbl 1085.35046
[35] Krupková, O.: The nonholonomic variational principle. J. Phys. A: Math. Theor. 42 2009 No. 185201 MR 2591195 | Zbl 1198.70008
[36] Krupková, O.: Noether Theorem, 90 years on. Geometry and Physics, Proc. XVII International Fall Workshop on Geometry and Physics, Castro Urdiales, Spain, 2008 , AIP Conf. Proceedings, American Institute of Physics, New York 2009 159–170
[37] Krupková, O.: Variational Equations on Manifolds. A.R. Baswell (ed.)Advances in Mathematics Research Vol. 9. Nova Science Publishers, New York 2009 201–274 MR 2599277
[38] Krupková, O., Musilová, J.: The relativistic particle as a mechanical systems with non-holonomic constraints. J. Phys. A: Math. Gen. 34 2001 3859–3875 DOI 10.1088/0305-4470/34/18/313 | MR 1840850
[39] Krupková, O., Musilová, J.: Non-holonomic variational systems. Reports on Math. Phys. 55 2005 211–220 DOI 10.1016/S0034-4877(05)80028-X
[40] Krupková, O., Prince, G.E.: Second Order Ordinary Differential Equations in Jet Bundles and the Inverse Problem of the Calculus of Variations. D. Krupka, D. Saunders (eds.)Handbook of Global Analysis Elsevier 2008 841–908 MR 2389647 | Zbl 1236.58027
[41] Krupková, O., Volný, P.: Euler-Lagrange and Hamilton equations for non-holonomic systems in field theory. J. Phys. A: Math. Gen. 38 2005 No. 8715 DOI 10.1088/0305-4470/38/40/015 | MR 2185849 | Zbl 1075.70021
[42] Krupková, O., Volný, P.: Differential equations with constraits in jet bundles: Lagrangian and Hamiltonian systems. Lobachevskii J. Math. 23 2006 95–150
[43] Massa, E., Pagani, E.: A new look at classical mechanics of constrained systems. Ann. Inst. Henri Poincaré 66 1997 1–36 MR 1434114 | Zbl 0878.70009
[44] Mestdag, T., Langerock, B.: A Lie algebroid framework for nonholonomic systems. J. Phys. A: Math. Gen. 38 2005 1097–1111 DOI 10.1088/0305-4470/38/5/011 | MR 2120934
[45] Noether, E.: Invariante Variationsprobleme. Nachr. kgl. Ges. Wiss. Göttingen, Math. Phys. Kl. 1918 235–257
[46] Sarlet, W.: A direct geometrical construction of the dynamics of non-holonomic Lagrangian systems. Extracta Mathematicae 11 1996 202–212 MR 1424757
[47] 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 10.1088/0305-4470/28/11/022 | MR 1344117 | Zbl 0858.70013
[48] Saunders, D.J.: The Geometry of Jet Bundles. London Math. Soc. Lecture Notes Series, Vol. 142. Cambridge Univ. Press, Cambridge 1989 MR 0989588 | Zbl 0665.58002
[49] Saunders, D.J., Sarlet, W., Cantrijn, F.: A geometrical framework for the study of non-holonomic Lagrangian systems II. J. Phys. A.: Math. Gen. 29 1996 4265–4274 DOI 10.1088/0305-4470/29/14/042 | MR 1406933 | Zbl 0900.70196
[50] Swaczyna, M.: Several examples of nonholonomic mechanical systems. Communications in Mathematics, to appear
[51] Volný, P., Krupková, O.: Hamilton equations for non-holonomic mechanical systems. O. Kowalski, D. Krupka, J. Slovák (eds.)Differential Geometry and Its Applications Proc. Conf., Opava, 2001. Mathematical Publications, Vol. 3. Silesian Univ., Opava, Czech Republic 2001 369–380 MR 1978791
Search
Partner of
