Electromagnetic Back-Action in Dipole–Loop Magnetomechanics: State–Space Model, Memory Kernel, and Jerk/Absement Asymptotes

A moving permanent magnet induces currents in nearby conductors and superconducting loops. This electrodynamic back-action produces forces that are history-dependent and, in general, cannot be represented by a conservative potential depending only on instantaneous separation. We derive a compact state--space model for the axial motion of a magnetic dipole coupled to an $N$-turn loop with inductance $L$ and resistance $R$. Using Faraday's law for the flux linkage $\Lambda(x)=N\Phi(x)$, the coupled dynamics are \[ L\dot i + R i = -\Lambda'(x)\dot x,\qquad F_{\mathrm{em}}(t)= i(t)\,\Lambda'(x(t)), \] which defines a passive, causal magnetomechanical memory element. Linearizing about an operating point $x_0$ yields an exact complex dynamic stiffness \[ K_{\mathrm{em}}(\omega) \equiv -\frac{F_{\mathrm{em}}(\omega)}{X(\omega)} = \frac{\Lambda'(x_0)^2\, i\omega}{R+i\omega L} = k_\infty\,\frac{i\omega\tau}{1+i\omega\tau}, \quad k_\infty=\frac{\Lambda'(x_0)^2}{L},\ \tau=\frac{L}{R}. \] We show that this back-action is exactly equivalent to a mechanical Maxwell element, derive closed-form expressions for added stiffness and added damping, and provide direct identification formulas for $(\tau,k_\infty)$ from measured complex stiffness. The dipole--loop geometry further admits an analytic design rule: the coupling gradient $|\Lambda'(x_0)|$ is maximized at $x_0=a/2$ where $a$ is the loop radius. Finally, we connect commonly proposed \emph{jerk-like} and \emph{absement-like} terms to controlled low- and high-frequency asymptotic expansions of the same passive kernel (with explicit validity limits). All predictions are validated by reproducible Python simulations, and code to generate figures and data is provided.