Cartan forms for first order constrained variational problems

Given a constrained variational problem on the 1-jet extension J1Y of a fibre bundle p:Y→X, under certain conditions on the constraint submanifold S⊂J1Y, we characterize the space of admissible infinitesimal variations of an admissible section s:X→Y as the image by a certain first order differential operator, Ps, of the space of sections Γ(X,s∗VY). In this way we obtain a constrained first variation formula for the Lagrangian density Lω on J1Y, which allows us to characterize critical sections of the problem as admissible sections s such that Ps+ELω(s)=0, where Ps+ is the adjoint operator of Ps and ELω(s) is the Euler-Lagrange operator of the Lagrangian density Lω as an unconstrained variational problem. We introduce a Cartan form on J2Y which we use to generalize the Cartan formalism and Noether theory of infinitesimal symmetries to the constrained variational problems under consideration. We study the relation of this theory with the Lagrange multiplier rule as well as the question of regularity in this framework. The theory is illustrated with several examples of geometrical and physical interest.