Utilization of Matrix Methods in Teaching Electrodynamics in Higher Education Institutions: Advantages and Perspectives
Authors: Belovol Oleksandr 
Published: May 2024

Problem. The instruction of physical and mathematical disciplines in higher education institutions employing cutting-edge methods and universal approaches, which seamlessly integrate scientific and engineering activities of future specialists, remains one of the pertinent and priority tasks. The utilization of multidimensional or even infinite-dimensional spaces has become an effective and nearly indispensable tool in mathematical modeling of physical phenomena. This circumstance is directly linked to the increasing level of abstraction and the refinement of mathematical methods. The use of geometric methods inherent in vector algebra and vector analysis as the foundation for studying mechanics, despite their illustrative nature, is losing its relevance. Methods built upon matrix formalism are evolving as substitutes. The matrix framework enables the exploitation of phase space advantages to derive canonical equations of motion for continuous media and electromagnetic field equations in covariant forms. The electromagnetic potential acquires mechanical significance, allowing the utilization of electromechanical analogies at a fundamental level rather than on a merely formal basis, as done in classical electrodynamics. Goal. The aim is to broaden the domain of matrix formalism application to infinite-dimensional spaces, substantiate its advantages compared to other approaches used in teaching the fundamentals of electrodynamics, and demonstrate its potential both in terms of modeling and utilizing computer technologies. Methodology. The methodological foundation for choosing matrix methods lies in the utilization of statistical modeling techniques for the motion of electromechanical systems in infinite-dimensional spaces. Additionally, electrical phenomena are viewed as manifestations of mechanical motion in spacetime. Such an approach enables the extensive use of analogical reasoning to interpret and understand equations that possess the most abstract nature. Results.  It has been demonstrated that the matrix method, previously applied in teaching classical mechanics, allows leveraging the benefits of transitioning to configuration space and phase space for obtaining canonical equations of relativistic mechanics and electromagnetic field equations. Moreover, the electromagnetic field is regarded as an integral part of relativistic dynamics. Originality. To enhance the understanding of electrodynamical processes, the mechanical nature of electrical phenomena is explored in this study, employing formal-mathematical analogies with classical mechanics. An inherent feature of the approach is the utilization of infinite-dimensional matrices with real coefficients, which, however, does not lead to significant complications. Practical value. The proposed method effectively formalizes the equations of electrodynamics, facilitating their utilization in solving practical problems with the aid of computer technologies, while remaining within the confines of standard mathematical preparation.