Conservation laws-a simple application to the telegraph equation

摘要

Conservation laws are a recognized tool in physical and engineering sciences. The classical procedure to construct conservation laws makes use of Noether's Theorem. It requires the existence of a Lagrange-function for the system under consideration. Two unknown sets of functions have to be determined. A broader class of such laws is obtained, if Noether's Theorem is applied together with the Bessel-Hagen extension, raising the number of sets of unknown functions to three. The same conservation laws can be obtained by using the Neutral-Action method with the advantage that only one set of unknown functions is required. Moreover, the Neutral-Action method is also applicable in the absence of a Lagrangian, since for this procedure only the governing differential equations are needed. By this, the Neutral-Action method appears to be the most useful tool in constructing conservation laws for systems with dissipation. The intention of this paper is to give a reference for further research on this topic rather than showing some detailed calculation on a special subject. Thus, the Neutral-Action method is introduced in detail and is applied to a simple example (the telegraph equation) to show the usefulness of this procedure for further applications in electronics.