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Nonholonomic variational systems. (English) Zbl 1134.37356
Summary: Nonholonomic mechanical systems on constraint submanifolds of first-order jet bundles are studied. A concept of variationality appropriate for constrained systems is introduced. The inverse problem for constraint motion equations is considered, and corresponding constraint variationality conditions, generalizing the Helmholtz conditions, are derived.

37J60 Nonholonomic dynamical systems
37K05 Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010)
70F25 Nonholonomic systems related to the dynamics of a system of particles
70F17 Inverse problems for systems of particles
Full Text: DOI
[1] Chetaev, N.G., On the Gauss principle, Izv. kazan. fiz.-mat. obsc., 6, 323-326, (1932-1933), (in Russian)
[2] Czudková, L.; Musilová, J., Variational non-holonomic systems in physics, (), in print · Zbl 1167.58010
[3] Giachetta, G., Jet methods in nonholonomic mechanics, J. math. phys., 33, 1652-1665, (1992) · Zbl 0758.70010
[4] Grácia, X.; Marín-Solano, J.; Muñoz-Lecanda, M.-C., Some geometric aspects of variational calculus in constrained systems, Rep. math. phys., 51, 127-148, (2003) · Zbl 1038.37052
[5] Helmholtz, H., Ueber die physikalische bedeutung des prinzips der kleinsten wirkung, J. für die reine u. angewandte math., 100, 137-166, (1887) · JFM 18.0941.01
[6] Krupka, D., Lepagean forms in higher order variational theory, (), 197-238 · Zbl 0572.58003
[7] Krupková, O., The geometry of ordinary variational equations, () · Zbl 1121.58020
[8] Krupková, O., Mechanical systems with non-holonomic constraints, J. math. phys., 38, 5098-5126, (1997) · Zbl 0926.70018
[9] Krupková, O., On the geometry of non-holonomic mechanical systems, (), 533-546 · Zbl 0937.37031
[10] Krupková, O., Recent results in the geometry of constrained systems, Rep. math. phys., 49, 269-278, (2002) · Zbl 1018.37041
[11] Krupková, O.; Musilová, J., Constraint Helmholtz conditions, (2002), Masaryk University Brno, 10 pp.; paper in preparation
[12] Krupková, O.; Swaczyna, M., The non-holonomic variational principle, (2002), Masaryk University Brno Brno, 34 pp.; paper to be published
[13] de León, M.; Marrero, J.C.; de Diego, D.M., Non-holonomic Lagrangian systems in jet manifolds, J. phys. A: math. gen., 30, 1167-1190, (1997) · Zbl 1001.70505
[14] Massa, E.; Pagani, E., Classical mechanic of non-holonomic systems: a geometric approach, Ann. inst. Henri Poincaré, 66, 1-36, (1997)
[15] Morando, P.; Vignolo, S., A geometric approach to constrained mechanical systems, symmetries and inverse problems, J. phys. A.: math. gen., 31, 8233-8245, (1998) · Zbl 0940.70008
[16] Sarlet, W., A direct geometrical construction of the dynamics of non-holonomic Lagrangian systems, Extracta mathematicae, 11, 202-212, (1996)
[17] Sarlet, W.; Cantrijn, F.; Saunders, D.J., A geometrical framework for the study of non-holonomic Lagrangian systems, J. phys. A: math. gen., 28, 3253-3268, (1995) · Zbl 0858.70013
[18] 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, 4265-4274, (1996) · Zbl 0900.70196
[19] Swaczyna, M., On the nonholonomic variational principle, (), in print · Zbl 1113.70016
[20] Saunders, D.J., The geometry of jet bundles, () · Zbl 0665.58002
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