On the inverse variational problem in nonholonomic mechanics. (English) Zbl 1271.49027
Summary: The inverse problem of the calculus of variations in a nonholonomic setting is studied. The concept of constrained variationality is introduced on the basis of a recently discovered nonholonomic variational principle. Variational properties of first order mechanical systems with general nonholonomic constraints are studied. It is shown that constrained variationality is equivalent to the existence of a closed representative in the class of 2-forms determining the nonholonomic system. Together with the recently found constrained Helmholtz conditions this result completes basic geometric properties of constrained variational systems. A few examples of constrained variational systems are discussed.
MSC:
| 49N45 | Inverse problems in optimal control |
| 58E30 | Variational principles in infinite-dimensional spaces |
| 70F25 | Nonholonomic systems related to the dynamics of a system of particles |
Keywords:
inverse problem of calculus of variations; Helmholtz conditions; nonholonomic constraints; nonholonomic variational principle; constrained Euler-Lagrange equations; constrained Helmholtz conditions; constrained Lagrangian; constrained ballistic motion; relativistic particleReferences:
| [1] | Bloch, A.M., Mestdag, O.E. Fernandez and T.: Hamiltonization of nonholonomic systems and the inverse problem of the calculus of variations. Reports on Math. Phys., 63, 2009, 225-249 · Zbl 1207.37045 · doi:10.1016/S0034-4877(09)90001-5 |
| [2] | Chetaev, N.G.: On the Gauss principle. Izv. Kazan. Fiz.-Mat. Obsc., 6, 1932-33, 323-326) |
| [3] | Helmholtz, H.: Ueber die physikalische Bedeutung des Prinzips der kleinsten Wirkung. J. für die reine u. angewandte Math., 100, 1887, 137-166 · JFM 18.0941.01 |
| [4] | Krupková, O.: Mechanical systems with non-holonomic constraints. J. Math. Phys., 38, 10, 1997, 5098-5126 · Zbl 0926.70018 · doi:10.1063/1.532196 |
| [5] | Krupková, O.: Recent results in the geometry of constrained systems. Reports on Math. Phys., 49, 2002, 269-278 · Zbl 1018.37041 · doi:10.1016/S0034-4877(02)80025-8 |
| [6] | Krupková, O.: The nonholonomic variational principle. J. Phys. A: Math. Theor., 42, 2009, 185201. · Zbl 1198.70008 · doi:10.1088/1751-8113/42/18/185201 |
| [7] | Krupková, O.: Geometric mechanics on nonholonomic submanifolds. Communications in Mathematics, 18, 1, 2010, 51-77 · Zbl 1248.70018 |
| [8] | Krupková, O., Musilová, J.: The relativistic particle as a mechanical system with non-holonomic constraints. J. Phys. A.: Math. Gen., 34, 2001, 3859-3875 · Zbl 1029.70008 · doi:10.1088/0305-4470/34/18/313 |
| [9] | Krupková, O., Musilová, J.: Nonholonomic variational systems. Reports on Math. Phys., 55, 2005, 211-220 · Zbl 1134.37356 · doi:10.1016/S0034-4877(05)80028-X |
| [10] | Krupková, O., Volná, J., Volný, P.: Constrained Lepage forms. Differential Geometry and its Applications, Proc. 10th Int. Conf. on Diff. Geom. and Appl., Olomouc, August 2007 (World Scientific, Singapore, 2008) 627–633 · Zbl 1167.58009 |
| [11] | Massa, E., Pagani, E.: Classical mechanic of non-holonomic systems: a geometric approach. Ann. Inst. Henry Poincaré, 66, 1997, 1-36 · Zbl 0878.70009 |
| [12] | Morando, P., Vignolo, S.: A geometric approach to constrained mechanical systems, symmetries and inverse problems. J. Phys. A.: Math. Gen., 31, 1998, 8233-8245 · Zbl 0940.70008 · doi:10.1088/0305-4470/31/40/015 |
| [13] | Sarlet, W.: A direct geometrical construction of the dynamics of non-holonomic Lagrangian systems. Extracta Mathematicae, 11, 1996, 202-212 |
| [14] | Swaczyna, M., Volný, P.: Uniform projectile motion as a nonholonomic system with a nonlinear constraint. Int. J. of Non-Linear Mechanics. Submitted · Zbl 1308.70017 |
| [15] | Tonti, E.: Variational formulation of nonlinear differential equations I, II. Bull. Acad. Roy. Belg. Cl. Sci., 55, 1969, 137-165, 262–278 · Zbl 0182.11402 |
| [16] | Vainberg, M.M.: Variational methods in the theory of nonlinear operators. 1959, GITL, Moscow) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
