Differential equations with constraints in jet bundles: Lagrangian and Hamiltonian systems

O. Krupková^a, P. Volný<sup>b

a</sup> 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.

Submitted by: V. V. Lychagin
Received: 24.07.2006

Language: English

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