Spectral and variational principles of electromagnetic field excitation in wave guides

Possible variational principles for excitation of an electromagnetic field in a wave guide are discussed. Our emphasis is not on the calculation of the modal shapes, which is common in previous art, but rather on the calculation of modal amplitude evolution, which are important in electron devices such as free electron lasers and gyrotrons. Variational principles have considerable importance in theoretical physics and are used among other things to derive numerical solution schemes, conservation laws via the Noether theorem and correct boundary conditions for the derived equations including the important effects of the backward waves amplitudes.