Dynamics of localized continuum inside a charged particle

This article explains how a charged body can retain its integrity despite electrostatic repulsion within itself. The study requires derivation of the field equations governing the velocity of any arbitrary element within the system. These relations are obtained from two Lagrangian principles based on obvious and least disruptive modifications of known concepts in classical mechanics. The two Lagrangians lead to a relativistic energy-momentum equation describing the motion and Maxwell's equations quantifying the electromagnetic fields, respectively. The energy-momentum relation shows that rigid body motions creating stationary fields in a rotating frame is a viable solution for the velocity inside the continuum. It also yields a combined potential which remains constant throughout the rotating particle. This constancy coupled with Maxwell's equations and boundary conditions for proper interfacial continuity provides the charge distribution and geometric shape required for continued coherence of the domain. The paper presents specific simulations for slender annuli steadily rotating about the axis of symmetry where the charge density and the axial width are plotted as radial functions. Curiously, when the analysis is extended to systems with different axes of rotation and symmetry, the proportionalities between energy and frequency as well as momentum and wave-number can be established. Such results are very similar to Planck's and de Broglie's laws, even though derived from classical principles. Thus, the presented theory might have deeper implications like, for example, in mathematical computation of fine structure constant.