Cartan algorithm and constraints in field theory

CAPRIOTTI, SANTIAGO

Madrid

Conferencia; The 10th AIMS Conference on Dynamical Systems, Differential Equations and Applications; 2014

AIMS, UAM-CSIC, ICMAT-CSIC

(Pre)multisymplectic manifolds provide a geometrical framework for the equations of motion of field theory, as (pre)symplectic manifolds does for dynamical systems. In the same way, there exists a procedure in the Hamiltonian formalism for field theories that allow us to deal with constraints, analogous to Gotay-Nester algorithm. The novel feature is that this algorithm has to deal with integrability issues, absent in the Gotay-Nester algorithm due to dimensional reasons; the natural outcome of this procedure is a set of decomposable multivectors, yielding to solutions of the Hamiltonian field equation whenever these integrability questions can be solved. On the other side, the Cartan algorithm is a procedure to ensure existence of integral manifold for linear Pfaffian exterior differential systems (EDS); namely, whether such algorithm can be applied to an EDS version of the Hamiltonian field equations, it is automatically ensured the question regarding integrability, and a set of constraints assuring integrability can be found. In the present talk we will show with some examples how to encode field theory and its equations in a linear Pfaffian EDS, and the way in which Cartan algorithm can be used in order to deal with them.