## 4 Conclusions  
The theory of gyro-devices comprises a hierarchy of physical models (analytical, linear, nonlinear, time-independent (static), time-dependent, and self-consistent physical models), culminating in the first-principles formulations (based on Maxwell equations and the relativistic dynamics of electrons in time-varying electromagnetic fields) used in advanced numerical particle-in-cell (PIC) codes.  
Over the years, these physical models have been implemented by many researchers from the gyrotron community in numerous stand-alone computer programs and software packages. We refer the reader to some of the most advanced and specialized numerical codes [74; 75; 76; 77; 78; 79; 80; 81; 82; 83]. In addition to CAD (computer-aided design) and optimization, these problem-oriented software packages are also used for numerical studies of the gyro-devices and for educational purposes. Our experience shows that analyzing the results of numerical experiments performed with different physical models is very informative for understand  
Figure 17: Trajectory analysis of EOS in conventional and LOG. Trajectory analysis of EOS in conventional (**a**) and LOG (**b**). Reprinted with the permission of IOP Publishing from Ref. [22].  
ing the underlying theory. Most of these modeling and simulation tools are in-house made codes that allow their users (who are also their developers) to modify the computational modules, input and output format, content, etc., as needed. However, some researchers (especially younger ones) nowadays prefer to use commercially available multiphysics software packages. We strongly discourage this, as we consider these to be, albeit powerful, \"black boxes\" (aka \"number crunching machines\") that cannot be modified. Based on our experience, we believe that one of the best ways to study physics (including gyrotron theory) is to program the physical models yourself. In any case, knowing the basics presented in this paper will facilitate the interpretation of the results of the numerical experiments.  
Although these fundamentals can be found scattered in numerous publications and monographs, they are not treated consistently and in sufficient detail. This was the main motivation for revisiting the most essential relations of gyrotron physics in an attempt (based on our teaching experience) to present them comprehensively in one place and in a proper order. We have endeavored to write a comprehensible, self-contained text that is understandable without resorting to additional sources. In this respect, the used carefully selected references should be considered mostly as suggestions for further reading for a more in-depth study of the discussed topics. We believe that the compendium of relationships and formulas presented here is essential for such studies and is aimed at the broad community of researchers working on the development and application of various gyro-devices. Furthermore, we hope that the material presented here in a pedagogical discourse will be a helpful advanced introduction to the physics of such devices. This research received no external funding. Not applicable. Not applicable. Not applicable. Data Availability Statement: The data presented in this study are available on request from the corresponding author. The data are not publicly available due to privacy.  
I pay tribute and express my gratitude to the late [PERSON], with whom we worked on some of the topics presented here. I also appreciate the curiosity and the questions of our students during the lectures and seminars at the Research Center for Development of Far-Infrared Region at the University of Fukui (FIR UF), Japan, that have always stimulated us to search for the best ways for teaching the physics of gyrotrons. The authors declare no conflict of interest.