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Lobachevskii Journal of Mathematics, 2006, Volume 23, Pages 95–150
(Mi ljm19)
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Differential equations with constraints in jet bundles: Lagrangian and Hamiltonian systems
O. Krupkováa, P. Volnýb a Palacký University
b VŠB – Technical University of Ostrava
Abstract:
The paper is a survey of the theory of Lagrangian systems with non-holonomic constraints in jet bundles. The subject of the paper are systems of second-order ordinary and partial differential equations that arise as extremals of variational functionals in fibered manifolds. A geometric setting for Euler–Lagrange and Hamilton equations, based on the concept of Lepage class is presented. A constraint is modeled in the underlying fibered manifold as a fibered submanifold endowed with a distribution (the canonical distribution). A constrained system is defined by means of a Lepage class on the constraint submanifold. Constrained EulerЧ-Lagrange equations and constrained Hamilton equations, and properties of the corresponding exterior differential systems, such as regularity, canonical form, or existence of a constraint Legendre transformation, are presented. The case of mechanics (ODEТs) and field theory (PDEТs) are investigated separately, however, stress is put on a unified exposition, so that a direct comparison of results and formulas is at hand.
Keywords:
jet bundles, non-holonomic constraints, semiholonomic constraints, holonomic constraints, constrained Lagrangian systems, constrained Euler–Lagrange equations, Hamilton–De Donder equations, regularity of constrained systems, momenta, Hamiltonian, Legendre transformation.
Citation:
O. Krupková, P. Volný, “Differential equations with constraints in jet bundles: Lagrangian and Hamiltonian systems”, Lobachevskii J. Math., 23 (2006), 95–150
Linking options:
https://www.mathnet.ru/eng/ljm19 https://www.mathnet.ru/eng/ljm/v23/p95
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Abstract page: | 340 | Full-text PDF : | 163 | References: | 36 |
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