# Spinning Coin

An interactive, entirely client-side simulator of a coin spun on a table: the rolling-disk
problem with its finite-time collapse. No account, no model call, no credits, no network
traffic after the page loads.

## What it does

- Integrates the exact equations of motion of a thin rigid disk rolling without slipping on a
  horizontal plane (McDonald & McDonald, arXiv:physics/0008227v3, eqs. 23-26, stated there to
  agree with Routh article 244), by classical RK4 with the step tied to the precession period.
- Tracks three quantities that are exactly conserved in the dissipationless limit: the energy,
  and a pair of first integrals obtained by eliminating time between two of the equations of
  motion, which leaves a linear system in the tilt angle. That pair is the classical
  integrability of the rolling disk (Appell, Korteweg, Chaplygin, Gallop).
- Offers the five published dissipation laws as one family, P = eps m R g Omega^beta:
  beta = 1 dry rolling friction; beta = 2 force linear in contact speed; beta = 2.5 Bildsten's
  boundary-layer air film; beta = 3 quadratic air drag; beta = 4 Moffatt's viscous air film.
- Measures the exponent live from the trajectory using the discriminator that needs no guess
  about the stopping time: dOmega/dt grows as Omega^(n+1).

## The claim it tests

"The coin spins down because of friction with the table, and the rattling speeds up because
it's losing energy."

The rattle rate is set by geometry alone: Omega^2 = g/(k R sin a) on the steady rolling family,
with no friction term in it. The energy on that family is E = (3/2) m g R sin a exactly, so
less energy does mean a faster rattle, but the causation runs through the tilt. And while the
coin rolls without slipping the contact force does no work at all, so "friction with the table"
has to mean rolling resistance, slipping or impacts - which is precisely the unsettled part.

## What it found

Every published exponent is derived by assuming the coin drifts along the steady family. Put
the same power law into the full equations and that assumption fails at a finite tilt:
alpha_turn scales as eps^(2/(beta+1)) and the peak precession as eps^(-1/(beta+1)). Measured
slopes agree with that derivation to better than 0.01 with r2 above 0.9998. Below that tilt the
precession stops rising and collapses: there is no singularity in the full dynamics at any
finite dissipation. The failure is earliest for the largest beta, which is Moffatt's air-film
law. A fitted exponent always comes back low, and the deficit shrinks as the dissipation weakens.

## Provenance

Every number printed on the page is generated from the offline harness output, not typed.
Sources, tags and the two errata found in the cited papers are listed on the page itself and in
CREDITS.txt. The literature name for the idealised object is a trademark of a commercial toy;
this app is named for the phenomenon and is unaffiliated with any product.