Invariant Reduction for Partial Differential Equations. II: The General Mechanism

Abstract: A mechanism of reduction of symmetry-invariant conservation laws, presymplectic structures, and variational principles of partial differential equations (PDEs) is proposed. The mechanism applies for an arbitrary PDE system that admits a local (point, contact, or higher) symmetry, and relates symmetry-invariant conservation laws, as well as presymplectic structures, variational principles, etc., to their analogs for systems that describe the corresponding invariant solutions. A version of Noether's theorem for the PDE system satisfied by symmetry-invariant solutions is presented. Several detailed examples, including cases of point and higher symmetry invariance, are considered.