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

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.