# How to calculate properties of a coupled electromagnetic / mechanical oscillator?

I'd like to study a class of systems which are (essentially) coupled electromagnetic/acoustic oscillators which can act as antennas for an electromagnetic field, but also can vibrate mechanically with resonant modes in a similar range as the electromagnetic field. I'm interested in calculating various properties like resonant frequency, Q factor, absorption/emission cross sections, magnitude of oscillations at equilibrium, etc. I'm not sure how simulate/calculate this, or what physics equations would be relevant. I'm interested in both simple back of the envelope equations, and suggestions for software for a more detailed simulation (FDTD, FEM, ...).

A simplified model system is a point dipole with a non-zero mass, attached to a 1D spring (with damping), within a constant dielectric medium which has incident electromagnetic plane waves. The particle would absorb energy from the field, vibrate, and re-radiate energy, as well as losing some energy due to mechanical damping. A slightly more complex model system would be a particle which is like a polarizable dipole: it has a fixed dipole moment, but external fields can increase that (something like a Drude particle).

The real system is pretty close to a spherical core-shell structure, where the core and shell have fixed charges (like an electret), and are made of polarizable dielectric materials with different densities and (anisotropic) elastic modulus. The shell may have multiple mechanical resonant modes (torsional, spheroidal, breathing etc). To give this some scale, the sizes are on the order of 100nm, the masses are femtograms, and the frequencies are in the GHz range.

To answer your first question, let's figure out the EOM for your simplified system. First, let's consider the system without damping. Electromagnetism is described by the following Lagrangian density \begin{align} \mathcal{L}_\text{EM} = -\frac14 F^2 + J A\,. \end{align} A simple harmonic oscillator is described by the Lagrangian \begin{align} L_\text{SHO} = \frac12m\dot X^2 - \frac12 k X^2\,. \end{align} We imagine that $$J$$ is a pure function of $$X - x$$ (where $$x$$ is the spatial coordinate). In this way, we have enforced that the charge density moves with the harmonic oscillator. Thus, the full Lagrangian is \begin{align} L = \int{\rm d} V\left(-\frac14 F_{\mu\nu}F^{\mu\nu} + A_\mu J^\mu(X - x)\right) + \frac12 m \dot X^2 - \frac12 k X^2 \end{align} The equations of motion are then (up to minus signs, I'll check these later...) \begin{align} \partial_\mu F^{\mu\nu} &= J^\nu\,,\\ \ddot X + \omega^2 X&=-\frac{1}{m}\int{\rm d} V A_\mu(\partial_x J^\mu)\,, \end{align} where $$\omega^2 = k/m$$. For example, if the charge density is just a point charge $$\rho = q\delta(X - x)\delta(y)\delta(z)$$, the curren density is just $$\rho \vec v$$, where $$\vec v$$ is the velocity of the charge. This can be expressed in terms of $$X$$ as $$\vec v = \dot X \hat x$$. Therefore, the equations of motion become \begin{align} \partial_\mu F^{\mu\nu} &= J^\nu\,,\\ \ddot X + \omega^2 X&=-\frac{q}{m}\int{\rm d} x \left(A_0\delta'(X - x) - A_x \dot X\delta(X - x)\right)\,,\\ &=-\frac{q}{m}\left(\partial_X A_0(t,X) - A_x(t,X) \dot X\right)\,. \end{align}