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Recent results in the geometry of constrained systems. (English) Zbl 1018.37041
The paper presents a nice, comprehensive survey of the author’s approach to the geometrical description of mechanical systems with nonholonomic constraints. This approach starts from the representation of an arbitrary system of second-order ordinary differential equations via a so-called dynamical form and its Lepage class on the second jet extension of a manifold fibred over a 1-dimensional base. The basic ingredients for such (unconstrained) dynamics are recalled in Section 2. Constraints are represented by a fibred submanifold \({\mathcal Q}\) of the first jet extension, leading to fundamental concepts such as the constraint ideal and the Chetaev bundle. An interesting result, quoted in Section 3, concerns the characterization of semi-holonomic constraints, for example by the property that the constraint ideal is a differential ideal. Semi-holonomic constraints turn out later (Section 6) to constitute the case in which the pullback of an unconstrained Lagrangian plays the role of Lagrangian for the constrained system. The main feature of this model is that constrained dynamical systems come from pulling back a representative of the Lepage class of a dynamical form to the constraint submanifold \({\mathcal Q}\). Constrained Lagrangian and Hamiltonian systems are discussed as particular cases within this general scheme.

MSC:
37J60 Nonholonomic dynamical systems
70F25 Nonholonomic systems related to the dynamics of a system of particles
58E30 Variational principles in infinite-dimensional spaces
70F17 Inverse problems for systems of particles
70H03 Lagrange’s equations
70H45 Constrained dynamics, Dirac’s theory of constraints
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[1] Cantrijn, F.; Sarlet, W.; Saunders, D.J., J. phys. A: math. gen., 32, 6869-6890, (1999)
[2] Chetaev, N.G., Izv. kazan. fiz.-mat. obsc., 6, 323-326, (1932-33)
[3] Giachetta, G., J. math. phys., 33, 1652-1665, (1992)
[4] Koon, W.S.; Marsden, J.E., Rep. math. phys., 40, 21-62, (1997)
[5] Krupková, O., The geometry of ordinary differential equations, () · Zbl 1121.58020
[6] Krupková, O., J. math. phys., 38, 5098-5126, (1997)
[7] Krupková, O., On the geometry of nonholonomic mechanical systems, (), 533-546 · Zbl 0937.37031
[8] Krupková, O., J. math. phys., 41, 5304-5324, (2000)
[9] Krupková, O., Differential systems in higher-order mechanics, (), 87-130 · Zbl 1034.70010
[10] Krupková, O.; Musilová, J., J. phys. A: math. gen., 34, 3859-3875, (2001) · Zbl 1029.70008
[11] O. Krupková and J. Musilová: to be published.
[12] de León, M.; Marrero, J.C.; de Diego, D.M., J. phys. A: math. gen., 30, 1167-1190, (1997)
[13] Massa, E.; Pagani, E., Ann. inst. Henri Poincaré, 55, 511-544, (1991)
[14] Massa, E.; Pagani, E., Ann. inst. Henri Poincaré, 66, 1-36, (1997)
[15] Sarlet, W., Extracta mathematicae, 11, 202-212, (1996)
[16] Sarlet, W.; Cantrijn, F.; Saunders, D.J., J. phys. A: math. gen., 28, 3253-3268, (1995) · Zbl 0858.70013
[17] Saunders, D.J.; Sarlet, W.; Cantrijn, F., J. phys. A: math. gen., 29, 4265-4274, (1996) · Zbl 0900.70196
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