Cartan forms for first order constrained variational problems

Abstract Given a constrained variational problem on the 1-jet extension J 1 Y of a fibre bundle p : Y → X , under certain conditions on the constraint submanifold S ⊂ J 1 Y , 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, P s , of the space of sections Γ ( X , s ∗ V Y ) . In this way we obtain a constrained first variation formula for the Lagrangian density L ω on J 1 Y , which allows us to characterize critical sections of the problem as admissible sections s such that P s + E L ω ( s ) = 0 , where P s + is the adjoint operator of P s and E L ω ( s ) is the Euler–Lagrange operator of the Lagrangian density L ω as an unconstrained variational problem. We introduce a Cartan form on J 2 Y 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.