Zöllner Extension

Research / Experimental. The Zöllner extension implements a 19th-century electrogravitational hypothesis. It is not part of standard Weber electrodynamics and is intended for comparative research, not general simulation. It is off by default.

Background

Johann Karl Friedrich Zöllner (1837–1882) proposed that matter consists of binary electrical dyads with a slight mismatch: the unlike-sign attraction is fractionally stronger than the like-sign repulsion. The residual attraction between neutral dyads produces an effective gravitational interaction.

The mismatch is parameterized by a > 0:

• Unlike-sign pairs: κᵢⱼ = 1 + a (stronger attraction)
• Like-sign pairs: κᵢⱼ = 1.0 (standard repulsion)

Each pair potential becomes:

U_ij = κ_ij · qᵢqⱼ / r · (1 - ṙ²/(2c²))

When a = 0 (or Zöllner disabled) all κ = 1 and the standard Weber potential is recovered exactly.

See Zöllner Electrogravitational Theory for the full theoretical derivation.

Enabling Zöllner

prob = HamiltonianProblem(
    sys, tspan, q0, p0;
    masses  = [1.0, 1.0],
    charges = [1.0, -1.0],
    c       = 10.0,
    dt      = 0.01,
    zollner = ZollnerOptions(enabled = true, a = 0.01),   # mismatch parameter; must be > 0
)

a must be strictly positive when enabled = true.

Combining with regularization

Zöllner is fully compatible with regularization. Zöllner is a problem-level option; regularization is an algorithm wrapper. Combine them by setting zollner on the problem and wrapping the algorithm:

prob = HamiltonianProblem(sys, tspan, q0, p0; ...
    zollner = ZollnerOptions(enabled = true, a = 0.01),
)

alg = RegularizedIntegrator(SymmetricProjectionIntegrator())
sol = solve(prob, alg)

κ values are automatically included in the parameter vector passed to regularization sub-steps, so the Zöllner coupling is respected even during close-encounter regularization. No extra configuration is needed.

Zöllner-specific plots

The Plots.jl extension provides four functions for visualizing the Zöllner contribution:

using Plots

energy = compute_energy_timeseries(sol)
forces = compute_pair_force_timeseries(sol, 1, 2)

plot_zollner_energy(energy)               # pair potentials + Zöllner extra
plot_zollner_force_residual(forces)       # force magnitude vs Zöllner fraction
plot_weber_vs_zollner(sol_weber, sol_z)   # overlay two solutions
plot_zollner_phase_space(f1, f2)          # (r, ṙ) portrait comparison

API reference

ZollnerOptions(; enabled=false, a=0.0)