From least action in electrodynamics to magnetomechanical energy -- a review

TL;DR

This review traces the derivation of electromechanical system equations from fundamental principles, explores magnetic energy concepts, and applies the theory to simple examples, clarifying properties of magnetic energy and Hamiltonian formalism.

Contribution

It provides a comprehensive review connecting Lagrangian and Hamiltonian formalisms to magnetomechanics, clarifying magnetic energy properties and gauge independence issues.

Findings

Hamiltonian formalism applies to electromechanical systems

Magnetic energy behaves as a typical potential energy at equilibrium

Examples illustrate the theory's application to real systems

Abstract

The equations of motion for electromechanical systems are traced back to the fundamental Lagrangian of particles and electromagnetic fields, via the Darwin Lagrangian. When dissipative forces can be neglected the systems are conservative and one can study them in a Hamiltonian formalism. The central concepts of generalized capacitance and inductance coefficients are introduced and explained. The problem of gauge independence of self-inductance is considered. Our main interest is in magnetomechanics, i.e. the study of systems where there is exchange between mechanical and magnetic energy. This throws light on the concept of magnetic energy, which according to the literature has confusing and peculiar properties. We apply the theory to a few simple examples: the extension of a circular current loop, the force between parallel wires, interacting circular current loops, and the rail gun. These show that the Hamiltonian, phase space, form of magnetic energy has the usual property that an equilibrium configuration corresponds to an energy minimum.