Orbit Reduction of Exterior Differential Systems, and group-invariant Variational Problems

For a given PDE system (or an exterior differential system) possessing a Lie group of internal symmetries the orbit reduction procedure is introduced. It is proved that the solutions of the reduced exterior differential system are in one-to-one correspondence with the moduli space of regular solutions of the prolongation of the original system. The isomorphism between the local characteristic cohomology of the reduced unconstrained jet space and the Lie algebra cohomology of the symmetry group is established. The group-invariant Euler-Lagrange equations of an invariant variational problem are described as a composition of the Euler-Lagrange operators on the reduced jet space and certain other differential operators on the reduced jet space. The practical algorithm of computing these operators is given.