Interactive companion to Joven (2026). Drag the bow. Watch the dual variable emerge.
Position along the string = radius from galactic center. Bow contact point = crossover radius (a = a₀). Subharmonic displacement maps to dark matter density.
The subharmonic emerges from two opposite corners of parameter space: slow velocity (understimulated) and high pressure (overwhelmed). Both enter the same narrow band. This heatmap shows the period ratio across the full parameter space. The crosshair tracks your current sliders.
Two entry paths to the same subharmonic.
Slow bow (understimulated): the string barely sustains oscillation,
period doubles as energy accumulates over two natural cycles.
High pressure (overwhelmed): the stick phase lengthens until the
string snaps free with enough force to skip a cycle.
Both converge to the same narrow band.
This is the MOND regime.
Dark matter effects don't appear in extreme gravitational fields
(black holes, neutron stars) or in negligible fields (empty space).
They appear in a specific low-acceleration band — the gravitational
analogue of the subharmonic regime shown in the heatmap.
The Stribeck friction curve, MOND interpolating function, and complementary slackness condition all produce the same threshold transition. Drag a₀ to see the mapping shift.
The dual variable oscillates around the saddle point. Slow step size → convergence (stable rotation curve). Fast step size → divergence (galaxy cluster regime).
The dark matter halo profile emerges as the shadow price of the rotation curve constraint. Where baryonic mass is insufficient, the dual variable fills the gap.
McGaugh et al. (2016). The empirical heart of the dark matter problem. The dual variable activates below a₀ — complementary slackness in action.
Companion to Stick-Slip Dynamics and the Dark Matter Dual (Joven, 2026). Notebook: stick_slip_lagrangian.ipynb. GitHub. Released under CC0.