From least action in electrodynamics to magnetomechanical energy—a review

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.

Magnetic Field and Current Are Zero Inside Ideal Conductors

“…According to a result by Greiner [32] a system will tend to maximize its magnetic energy when the conductor geometry changes while currents are kept constant. This has also been discussed in Essén [26]. In conclusion, if currents are kept constant the magnetic energy (A10) will tend (thermodynamically) to a stable equilibrium with at a maximum value and we note that this corresponds to a pure surface current ξ = 1.…”

“…There will be forces on a loop of current that encloses a magnetic flux that expands it [5]. This is the well know mechanism behind the rail gun, see, e.g., Essén [26]. All the little current loops in the metal will thus expand until they come to the surface where the expansion stops.…”

Algebraic motion of vertically displacing plasmas

“…Our analysis can be extended to more elaborate wall geometries by using a Lagrangian principle in order to encode the inductive coupling between multiple wall pieces and the moving plasma column. To include resistive effects, the velocity gradient of the so-called Rayleigh dissipation function is added to the Euler-Lagrange equations [28]. This framework extends the formulation of engineering codes such as VALEN [29], commonly used to compute wall Eddy currents for the design of vacuum vessels, to self-consistently include the effect of a vertically displacing plasma.…”

“…δ = ∆/a is the normalised diameter of the coils, = a/R the inverse aspect ratio and L c = µ 0 l/2π = µ 0 R. The normalised mutual inductance,M (z) = M (z)/L c where L c = µ 0 l/2π, of two circular coils of equal major radius R = a/ is given as a function of the separating normalised vertical distancez = z/a by[28] m sin 2 θdθ the complete elliptic integral of second kind, /R is the inverse aspect ratio.It is noted that the argument of the elliptic functions is bounded by 2/ √ 5 ∼ = 0.89 ≤ k ≤ 1. Due to the logarithmic divergence of K(k) as k → 1, the mutual inductance of two circular coils becomes well approximated bȳ M (z) −→ − ln( z) (B4)…”

Faraday's Law and Magnetic Induction: Cause and Effect

“…How the charges as a whole respond to an applied force is, from a microscopic perspective, a complicated matter. It involves how the charges individually experience the local environment of the conductor, but also how they interact with each other[15,21,22]. Here we simply reduce the combined effect of all such complications down into an effective mass density 4.…”