Symmetry and the Vector Helmholtz Equation in Polar Coordinates.

The vector Helmholtz equation arises in the study of electromagnetism and elastodynamic wave propagation. The conventional method for solving the vector Helmholtz equation requires that the solution be decomposed into the sum of a gradient of a scalar potential and the curl of a vector potential. We present here an alternative procedure for finding the solution that uses the symmetry group of the vector Helmholtz equation. This new technique is related to the classical separation of variables approach to solving the scalar Helmholtz equation and reduces to scalar separation of variables in the case of one component vector fields. We show how the vector Helmholtz equation can be solved in polar coordinates using this method.