Conservation Laws of Uniform Linear Homogeneous Systems

Abstract

A generalized linear homogeneous system is analyzed for the quadratic forms that are invariant under it. Such forms lead to conservation laws that are properties of the system. If the system has n degrees of freedom, it will have at least n such conservation laws which together determine the ``type'' of the system. The analysis may be viewed as the determination of the metrics under which the operator becomes rotational. In such a metric, the square of the length of a vector is invariant. A conservation law is expressible as its appropriate metric. The Manley‐Rowe relations for parametric devices, and Chu's kinetic power theorem for electron‐beam devices are examples. The analysis is related both to the system operator and to a possible system matrix differential equation. In the latter case, this allows a direct connection between the conservation laws and the physical construction of the system. The converse problem in which two metrics are specified is also considered. The range of possible matrix differential equations of a given type, and of system operators, is determined. Application is made to a uniform parametric line with ``ideal forward coupling.'' Such a device will be stable in the sense that it avoids internal feedback. If it is matched either on the input or output side, it will not oscillate regardless of its gain or electrical length. The necessary and sufficient conditions for this ideal behavior are obtained by considering the invariant forms—the conservation laws—that it should obey.