Field-Theory Analogs of the Lagrange and Poincaré Invariants

Abstract

The Lagrange differential invariant and the Poincaré integral invariant of classical dynamics have as their analogs in Lagrangian field theory a ``differential divergence-free vector'' and an ``integral divergence-free vector.'' The former, which is expressible as a divergence-free vector-bracket expression, may be used to derive conservation relations associated with the transformation properties of a given system. It is not necessary that these transformations should be infinitesimal; by way of example, conservation theorems are established for systems which are periodic and for systems which are invariant under spatial inversion. The differential divergence-free vector may also be used to establish reciprocity and orthogonality relations: simple examples which are here discussed are Betti's reciprocal theorem of elasticity and Lorentz's reciprocal relation of electromagnetic theory. An extended form of the differential divergence-free vector allows for variation not only of the dependent variables but also of the independent variables. The integral divergence-free vector associates a conserved quantity with any closed one-parameter family of solutions of the field equations. As examples, we derive the ``equation of conservation of probability'' of quantum mechanics, and a classical form of the relation between the momentum and wave vectors for a plane wave in a propagating medium. The theorem of classical dynamics relating a complete set of Poisson brackets to a complete set of Lagrange brackets cannot be extended to the present formalism. The formula which represents the obvious extension of the classical formula for the Poisson bracket is of no interest since it can be shown not to be canonically invariant.