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Abstract The calculation of the gravitational self-energy of a disk with constant surface mass density is crucial because it quantifies the energy required to assemble the disk from individual particles considering their mutual gravitational attraction. This concept is important in astrophysics and planetary science, especially when analyzing the formation and stability of structures like galaxies or accretion disks around black holes. In this work, we show how to calculate the gravitational self-energy of a disk with constant surface mass density by using Fourier transform techniques. From a pedagogical perspective, finding the gravitational self-energy of a disk using the Fourier transform method helps undergraduate students learn how to use powerful mathematical tools to solve physics problems more easily. It also shows how symmetry in combination with the Fourier transform method can simplify the calculation of complicated multi-dimensional integrals that arise in common topics in classical mechanics. 

Abstract We study the classical motion of a charged particle in presence of an inductively increasing time-dependent magnetic field as the one created inside a resistor-inductor series circuit driven by a voltage source. The inductor is treated as an infinite solenoid. In such a scenario, the expression for the time-dependent magnetic field generated when the circuit is turned on can be easily derived. We consider the case study of two-dimensional motion since the generalization to three-dimensions is elementary. The resulting differential equations for the two-dimensional motion of the charged particle are solved by using a particular method which relies in deployment of complex variables. The ensuing motion has interesting features that highlight the challenges faced in studies of charged particles in a time-dependent magnetic field. This study has applications in magnetic plasma confinement, where understanding charged particle dynamics in time-varying magnetic fields helps optimize stability and energy retention in fusion devices. 

Structures composed of classical dipoles in higher-dimensional space present a unique opportunity to venture beyond the conventional paradigm of few-body or cluster physics. In this work, we consider the six convex regular polychora that exist in an Euclidean four-dimensional space as a theoretical benchmark for hte investigation of dipolar systems in higher dimensions. The structures under consideration represent the four-dimensional counterparts of the well-known Platonic solids in three-dimensions. A dipole is placed in each vertex of the structure and is allowed to interact with the rest of the system via the usual dipole–dipole interaction generalized to the higher dimension. We use numerical tools to minimize the total interaction energy of the systems and observe that all six structures represent dipole clusters with a zero net dipole moment. The minimum energy is achieved for dipoles arranging themselves with orientations whose angles are commensurate or irrational fractions of the number π.