Theory primer

This page distills the physics encoded in the LW Integrator and mirrors the notation used in the internal design notes that accompany the project. It connects the covariant Liénard–Wiechert formalism to the concrete data structures exposed in core/trajectory_integrator.py and the validation studies under examples/validation.

Note that throughout the codebase Gaussian units are used. The unorthodox choice of amu-millimeter-nanosecond units is implemented to avoid numerical overflows.

Retarded fields

The solver models every source particle as a point charge whose fields are sampled at the observer’s retarded time. Starting from Jackson’s form of the Liénard–Wiechert fields, the magnetic field is obtained from the electric field via a cross product, while the electric field splits into a velocity term and an acceleration term:

\[\mathbf{B} = \bigl[\mathbf{n} \times \mathbf{E}\bigr]_{\text{ret}},\]
\[\mathbf{E} = e\left[\frac{\mathbf{n} - \boldsymbol{\beta}}{\gamma^{2}\,\kappa^{3} R^{2}}\right]_{\text{ret}} + \frac{e}{c} \left[ \frac{\mathbf{n} \times \bigl((\mathbf{n} - \boldsymbol{\beta}) \times \dot{\boldsymbol{\beta}}\bigr)}{\kappa^{3} R} \right]_{\text{ret}},\]

where \(\kappa = 1 - \boldsymbol{\beta} \cdot \mathbf{n}\), \(R\) is the retarded source–observer separation, \(\boldsymbol{\beta} = \mathbf{v}/c\), and \(\gamma = (1-\beta^{2})^{-1/2}\). Each quantity is evaluated at the retarded time \(t - R/c\). The implementation samples these terms inside core.trajectory_integrator.retarded_integrator(), looping over all available source trajectories.

A key limit for the benchmark problems is the near head-on configuration where \(\mathbf{n}\) aligns with \(\boldsymbol{\beta}\). Neglecting transverse components yields

(1)\[\mathbf{E}_{\parallel} \approx e\,\frac{1-\beta}{(1+\beta) R^{2}}\,\mathbf{n},\]

which explains the asymptotic growth of the longitudinal field as \(\beta \rightarrow 1\). This is the regime probed by the aperture-loss studies and the recoil-reduction scenarios in examples/validation.

Covariant potentials

Instead of tracking fields directly, the integrator evolves the covariant potential \(A^{\alpha}\) for each source trajectory. Using proper time \(\tau\) as the integration variable, the retarded potential reads

(2)\[A^{\alpha}(x) = \left.\frac{e\, V^{\alpha}(\tau)}{V(\tau) \cdot [x - r(\tau)]}\right|_{\tau = \tau_{0}},\]

with \(V^{\alpha} = \{c\gamma, \gamma \mathbf{u}\}\) the four-velocity, \(r^{\alpha}(\tau)\) the source worldline, and \(\tau_{0}\) obtained from light-cone constraint \([x - r(\tau_{0})]^{2} = 0\). The denominator reduces to \(\gamma c R \kappa\), linking the potential back to the geometry used in (1).

Conjugate momentum and equations of motion

Each observer particle carries a conjugate four-momentum

(3)\[\mathcal{P}^{\alpha} = m V^{\alpha} + \frac{e}{c} A^{\alpha},\]

where \(m\) and \(e\) are the observer mass and charge. Differentiating \(\mathcal{P}^{\alpha}\) with respect to proper time leads to the mixed-field force law used inside the stepping kernel:

(4)\[\frac{d\mathcal{P}^{\alpha}}{d\tau} = \frac{e}{c} V_{\beta} \, \partial^{\alpha} A^{\beta}.\]

Expanding \(\partial^{\alpha} A^{\beta}\) in terms of \(V^{\alpha}\), \(R^{\alpha}\), \(\dot{V}^{\alpha}\), and \(\kappa\) yields the component-wise form implemented in core.trajectory_integrator._update_conjugate_momentum. The spatial components couple velocity, acceleration, and retarded distance, ensuring that head-on image-charge interactions reproduce the steep gradients reported in the reference study.

Position updates follow directly from the Hamiltonian identity

(5)\[\frac{d x^{\alpha}}{d\tau} = \frac{1}{m}\left( \mathcal{P}^{\alpha} - \frac{e}{c} A^{\alpha} \right),\]

which the solver evaluates after each momentum update to keep particle states in sync. Proper-time stepping avoids runaway behaviour at high \(\gamma\) while keeping the integration scheme close to the legacy implementation (see legacy/covariant_integrator_library.py for a verbatim reference).

Radiation pressure and reaction

The validation notebooks explore scenarios where residual fields act on a test particle once a conducting surface or driving bunch is withdrawn. Two secondary forces are monitored to confirm that their contribution is negligible for the reported configurations:

  • Radiation pressure. Using Jackson’s scaling, the momentum transfer to an

    observer with area \(a_{T}\) receiving power \(P_{R}\) across solid angle \(\Omega\) is \(\dot{P}_{\text{RP}} = (P_{R}/c)\,(a_{T}/\Omega R^{2})\). For the millimetre-to-micron geometries in this repository, this quantity is orders of magnitude smaller than the Lorentz force recovered from (1).

  • Radiation reaction. Medina’s reduced-order form of the Lorentz–Abraham–Dirac force is used to damp numerical instabilities near conducting boundaries:

    \[\mathbf{F}_{\text{rad}} = \frac{2}{3}\frac{e^{2}}{m c^{3}}\left[\frac{d\gamma}{dt}\,\mathbf{F}_{\text{ext}} - \frac{\gamma^{3}}{c^{2}} (\mathbf{F}_{\text{ext}} \cdot \mathbf{a})\, \mathbf{v}\right].\]

    The implementation only activates this term when image-charge interactions drive \(R\) toward the micron scale so that the retarded integrator can report a stable pre-impact energy.

Bridging back to the code

The mathematical relationships above surface in the codebase as follows:

For deeper derivations and experimental context, see the technical note in LW_local_refs/main.tex.