Losslessness in Nonlinear Kirchhoff Circuits

and in Relativity Theory

Alfred Fettweis

Ruhr-Universität Bochum

D-44780 Bochum, Germany

Kirchhoff circuits are of importance not only for studying electrical phenomena but are ide-ally 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 essential. They consist of interconnections of a variety of elements, which nowhere have to be linear and con-stant. If an element such as an inductance is nonlinear and/or explicitly time dependent and is to be characterized as being passive or, more specifically, lossless, its defining relation must have a specific form, but the classical relation for a relativistic mass is not of this type. It is shown that, preserving classical relativistic kinematics, requiring relativistic dynamics to ap-proach the Newtonian one for appropriate limits, and putting prime emphasis on work done, thus on 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 in-ductance. This alterna-tive 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 contradict it, as implausible as that theory may a priori appear to be. The paper complements and updates earlier results on the subject and improves the presentation.