Nonlinear Kirchhoff Circuits and Relativity Theory

Abstract

Kirchhoff circuits consist of interconnections of elements. They are of importance not only for studying electrical phenomena but are ideally suited to model a broad range of physical systems for purposes where conservation of power and energy and related concepts such as passivity and losslessness are of paramount importance. In order to properly characterize such properties in the nonlinear case, the defining relations for nonlinear inductances etc. must have a specific form, but the classical relation for a relativistic mass is not of this type. It is shown that, preserving the classical relativistic kinematics and imposing a very reasonable requirement concerning work done, thus energy rather than momentum, one is naturally led to an expression for force in terms of mass and velocity whose form is in full agreement with that referred to for a nonlinear inductance. This alternative way of modifying Newton's second law requires Newton's third law to be also modified. These two modifications combined produce the same conservation of momentum and the same dynamics of particles in fields as classical relativity. The expression for kinetic energy, however, is different. Logically consistent derivations are presented, and a theoretical and an experimental result are pointed out that tend to offer some support to the alternative theory, or at least do not outrightly contradict it, as implausible as that theory may a priori appear to be. The paper updates earlier results on the subject.