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Various approaches to conservative and nonconservative nonholonomic systems. (English) Zbl 0931.37023

Two problems are the main contents of this paper. The first one is the problem of so-called ‘unnatural’ constraints, which are obtained by a limiting process and are nonlinear in the velocities. For these constraints the generalized d’Alembert principle is not valid anymore. The second one is the reduction problem for mechanical systems with nonholonomic constraints.
The author proposes a geometric setting for the Hamiltonian description of these systems and compares his approach where he considered only time-independent constraints to the known ones. The realization of these possibly nonconservative systems can now be found in sensors, servomechanisms and other high tech devices. The kinematic properties of the constraint are described by the Hamiltonian constraint submanifold of the phase space whereas the dynamical properties by a vector subbundle of the tangent bundle to the phase space along the Hamiltonian constraint submanifold. The author generalizes the setting by using a Poisson structure on the phase space instead of the canonical symplectic structure of a cotangent bundle. So a very straightforward reduction procedure can now be applied.

MSC:

37J60 Nonholonomic dynamical systems
37K05 Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010)
70H33 Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction for problems in Hamiltonian and Lagrangian mechanics
37J15 Symmetries, invariants, invariant manifolds, momentum maps, reduction (MSC2010)
70F25 Nonholonomic systems related to the dynamics of a system of particles
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[1] Appell, P., C. R. Acad. Sc. Paris, 152, 1197-1199 (1911)
[2] Appell, P., Rend. Circ. Mat. Palermo, 32, 48-50 (1911)
[3] Arnol’d, V. I.; Kozlov, V. V.; Neishtadt, A. I., Mathematical aspects of classical and celestial mechanics, (Arnol’d, V. I., Dynamical Systems III (1988), Springer: Springer Berlin, Heidelberg, New York) · Zbl 0674.70003
[4] Barone, F.; Grassini, R.; Mendella, G., A unified approach to constrained mechanical systems as implicit differential equations (1996), Università di Napoli, preprint · Zbl 0965.70030
[5] Bates, L.; Śniatycki, J., Rep. Math. Phys., 32, 99-115 (1991)
[6] Bates, L.; Graumann, H.; MacDonnell, C., Rep. Math. Phys., 37, 295-308 (1996)
[7] Benenti, S., Geometrical aspects of the dynamics of non-holonomic systems, Journées relativistes, Chambéry (14-16 May 1987)
[8] Bloch, A. M.; Krishnaprasad, P. S.; Marsden, J. E.; de Alvarez, G. Sánchez, Automatica, 28, 745-756 (1992)
[9] Bloch, A.; Krishnaprasad, P. S.; Marsden, J. E.; Murray, R. M., Arch. Ration. Mech. Anal., 136, 21-99 (1996)
[10] Cardin, F.; Favretti, M., On nonholonomic and vakonomic dynamics of mechanical systems with nonintegrable constraints, J. Geom. Phys., 18, 295-325 (1996) · Zbl 0864.70007
[11] Cariñena, J. F.; Rañada, M. F., J. Phys. A: Math. Gen., 26, 1335-1351 (1993) · Zbl 0772.58016
[12] Chetaev, N. G., On Gauss principle, Izv. Kazan Fiz. Mat. Obs., 6, 323-326 (1932), (in Russian)
[13] Cushman, R.; Kemppainen, D.; Śniatycki, J.; Bates, L., Rep. Math. Phys., 36, 275-286 (1995)
[14] Cushman, R.; Hermans, J.; Kemppainen, D., The rolling disc, Progress in nonlinear differential equations and their applications, 19, 21-60 (1996) · Zbl 0849.58028
[15] Dazord, P., Illinois J. Math., 38, 1, 148-175 (1994)
[16] Delassus, E., C. R. Acad. Sc. Paris, 152, 1739-1743 (1911)
[17] Delassus, E., C. R. Acad. Sc. Paris, 153, 626-628 (1911)
[18] de León, M.; de Diego, D. Martín, J. Math. Phys., 37, 7, 1-26 (1996)
[19] de León, M.; J. C. Marrero, M.; de Diego, D. Martín, Mechanical systems with non-linear constraints (1996), preprint
[20] Hermans, J., Nonlinearity, 8, 4, 493-515 (1995)
[21] Hermans, J., Rolling rigid bodies with and without symmetries, (Ph.D. thesis (1995), University of Utrecht)
[22] Ibort, A.; de León, M.; Marmo, G.; de Diego, D. Martín, Non-holonomic constrained systems as implicit differential equations, (Proceedings of the Workshop on Geometry and Physics in honour of W. Tulczyjew. Proceedings of the Workshop on Geometry and Physics in honour of W. Tulczyjew, Vietri, Italy (September 29-October 3, 1996)), to appear in the · Zbl 0928.70031
[23] Koiller, J., Arch. Ration. Mech. Anal., 118, 133-148 (1992)
[24] Koon, W.-S.; Marsden, J. E., SIAM J. Control Optim., 35, 901-929 (1997) · Zbl 0880.70020
[25] Koon, W.-S.; Marsden, J. E., Poisson reduction for nonholonomic mechanical systems with symmetry, (Preprint of the California Institute of Technology (1997)) · Zbl 1120.37314
[26] Korteweg, D. J., Über eine ziemlich verbreitete unrichtige Behandlungssweise eines Problemes der rollen-den Bewegung, über die Theorie dieser Bewegung, und insbesondere über kleine Rollende Schwingungen um eine Gleichgewichtslage, Nieu Archief voor Wiskunde, 4, 130-155 (1899) · JFM 30.0639.02
[27] Lewis, A. D.; Murray, R. M., Int. J. Non-Linear Mechanics, 30, 793-815 (1995)
[28] Lewis, A. D., Rep. Math. Phys., 38, 11-28 (1996)
[29] Lichnerowicz, A., J. Differential Geometry, 12, 253-300 (1977)
[30] Marle, C.-M., Commun. Math. Phys., 174, 295-318 (1995)
[31] Marle, C.-M., Kinematic and geometric constraints, servomechanisms and control of mechanical systems, (Proceedings of the Workshop on Geometry and Physics in honour of W. Tulczyjew. Proceedings of the Workshop on Geometry and Physics in honour of W. Tulczyjew, Vietri, Italy (September 29-October 3, 1996)), to appear in the · Zbl 0915.70018
[32] Marmo, G.; Mendella, G.; Tulczyjew, W. M., Constrained Hamiltonian systems as implicit differential equations (1996), preprint · Zbl 0766.58020
[33] Massa, E.; Pagani, E., Ann. Inst. Henri Poincaré, Physique théorique, 55, 1, 511-544 (1991)
[34] Massa, E.; Pagani, E., Ann. Inst. Henri Poincaré, Physique théorique, 66, 1, 1-36 (1997)
[35] Pironneau, Y., Sur les liaisons non holonomes non linéaires, déplacements virtuels à travail nul, conditions de Chetaev, (Proceedings of the IUTAM-ISIMM Symposium on Modern Developments in Analytical Mechanics, vol. II (1983), Accademia delle Scienze di Torino: Accademia delle Scienze di Torino Torino), 671-686, Torino, June 7-11, 1982
[36] Sarlet, W.; Cantrijn, F.; Saunders, D. J., J. Phys. A: Math. Gen., 28, 3253-3268 (1995) · Zbl 0858.70013
[37] Tulczyjew, W. M., C. R. Acad. Sc. Paris, 280, A, 1295-1298 (1975)
[38] Tulczyjew, W. M., Ann. Inst. Henri Poincaré, 27, 101-114 (1977)
[39] Tulczyjew, W. M., Geometric formulations of physical theories (1989), Bibliopolis: Bibliopolis Napoli, Italy · Zbl 0707.58001
[40] Van der Schaft, A. J.; Maschke, B. M., Rep. Math. Phys., 34, 225-233 (1994)
[41] Vershik, A. M., Classical and non-classical dynamics with constraints, (Borisovich, Yu. G.; Gliklikh, Yu. E., Global Analysis, Studies and Applications 1. Global Analysis, Studies and Applications 1, Lecture Notes in Mathematics, 1108 (1984), Springer: Springer Berlin), 278-301 · Zbl 0554.58024
[42] Vershik, A. M.; Faddeev, L. D., Sci. Math. Sov., 1, 339-350 (1981)
[43] Vershik, A. M.; Gershkovich, T. Ya., Nonholonomic dynamical systems, geometry of distributions and variational problems, (Arnold, V. I.; Novikov, S. P., Dynamical Systems VII (1994), Springer: Springer Berlin), 1-81
[44] Weber, R. W., Arch. Ration. Mech. Anal., 92, 309-335 (1985)
[45] Weinstein, A., J. Differential Geometry, 22, 255 (1985)
[46] Whittaker, E. T., A Treatise on the Analytical Dynamics of Particles and Rigid Bodies (1937), Cambridge University Press, reissued with Foreword · Zbl 0061.41806
[47] Zenkov, D. V., J. Nonlinear Sci., 5, 503-519 (1995)
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