Mathematical Model of Permanent Magnets and Superconducting Coils

A Lagrangian formalism is developed, which allows finding the magnetic potential energy of interactions in a system of superconducting inductors (coils with constant magnetic flux) and permanent magnets (coils with direct current). The explicit form of potential energy for the magnetic system with constant magnetic fluxes and direct currents allows the stability analysis in such magnetic systems both at equilibrium and in motion. A number of applications are indicated, which will benefit from this approach such as modeling of cyber-physical or technical systems of magnetic levitation.