Differential equations with constraints in jet bundles: Lagrangian and Hamiltonian systems
The authors present a survey of the theory of Lagrangian systems with non-holonomic constraints in jet bundles. The exposition deals with systems of second-order ordinary and partial differential equations that arise as extremals of variational functionals in fibered manifolds. A geometrical 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 Euler-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 cases of mechanics (ODEs) and field theory (PDEs) are investigated separately; still, the exposition is unitary, which permits a direct comparison of results and formulas. It should be noted that, although the paper is of review type, it contains new results regarding constrained Hamilton-De Donder equations and constraint Legendre transformation for general non-holonomic constraints in field theory.
scientific article; zbMATH DE number 5141815