Vector potentials for symmetric hyperbolic systems

Abstract

0. Introduction. Potentials are widely used in the physics literature in the study of a great variety of wave-propagation problems, but it is not altogether clear how (or if) a potential for a particular problem might be discovered. In a recent paper [4] J. Schulenberger showed that the Lagrange identities and for Maxwell's equations and the equations of two-dimensional elasticity derive in a natural way from the spectral resolutions of the corresponding operators. More recently, J. Schulenberger and the author have shown [3] that there is a class of symmetric hyperbolic systems, describing most wave-propagation phenomena of classical physics, which is characterized by admitting of a special form, called there potentials of classical type. These are very useful in studying symmetries, degeneracies, and conserved quadratic forms, but they suffer the disadvantage of being nonlocal. It is the purpose of this paper to show that this same class of symmetric hyperbolic systems is also characterized by another type of potential decomposition which is very often in applications local and which even more strongly emphasizes the special nature of this class of equations. It is, furthermore, easier to compute these local than those of [3](cf. §3).