Spinning Coin — credits, provenance and what differs
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WHAT THIS IS

An independent, from-scratch simulator of a physical phenomenon: a coin spun flat on a table,
which rattles faster and faster as it lies down and then stops abruptly. It is not a
reimplementation of any product or of anyone else's software. No code, art, sound or text was
taken from another program.

THE NAME

The physics literature calls the idealised object "Euler's disk". That name is also a
trademark: a commercial scientific demonstration toy, invented by Joseph Bendik between 1987
and 1990 and named by him after Leonhard Euler, distributed by Tangent Toy Co. This app is
therefore named for the phenomenon — a spinning coin — and the literature name appears only
where a paper is being cited. Spinning Coin is not affiliated with, endorsed by, sponsored by
or derived from Tangent Toy Co. or any other maker, and it does not replicate any product.

WHOSE PHYSICS

The equations of motion, the unified dissipation family and the lift-off criterion are taken
from published work and cited on the page:

  A. J. McDonald & K. T. McDonald, "The Rolling Motion of a Disk on a Horizontal Plane",
    arXiv:physics/0008227v3 (2002). Equations 23-26 (motion and energy), 29 (steady motion),
    75-81 (the P = eps m R g Omega^beta family, the exponents, and the lift-off cut-off),
    85 (the t0-free discriminator). They state their equations agree with Routh, article 244.
  H. K. Moffatt, "Euler's disk and its finite-time singularity", Nature 404, 833-834 (2000).
    The air-film model, alpha = (2 pi eta R / m)^(1/3) (t0-t)^(1/3), and the lift-off time.
  G. van den Engh, P. Nelson & J. Roach, "Analytical dynamics: numismatic gyrations",
    Nature 408, 540 (2000), with Moffatt's reply in the same issue. The vacuum experiment.
  K. Easwar, F. Rouyer & N. Menon, "Speeding to a stop: the finite-time singularity of a
    spinning disk", Phys. Rev. E 66, 045102 (2002). The measured n = 2.7 to 3.2.
  L. Bildsten (2002), via the citations above: the boundary-layer revision, exponent 4/9.
  T. Baranyai & P. L. Varkonyi, "Imperfections, impacts, and the singularity of Euler's disk",
    arXiv:1706.10205 (2017). The impact channel, and the review of Leine's conclusions.
  T. Kilty, "Re: Euler's Disc and its finite-time singularity", kilty.com/euler.htm.
  Wikipedia, "Euler's Disk", for the toy's inventor, trademark status and dimensions.
  US Mint and euro coin specifications, for the coin presets.

The Nature articles are paywalled. Moffatt's scaling law and lift-off time are quoted from two
independent secondary sources that agree with each other, and the harness checks that
Moffatt's published lift-off time and McDonald & McDonald's general formula give the same
number at beta = 4 — they agree to the last bit of a double.

WHAT IS ORIGINAL HERE

  - All code: engine, renderer, interface, harness, image generation.
  - The construction of the two extra exact invariants as a numerical transport along the
    linear system in the tilt angle, and their verification on the full nonlinear trajectories.
  - The injection of the published power-law dissipation into the full equations of motion as
    an energy sink with a selectable channel. The published derivations use the adiabatic
    reduction instead; this is a modelling choice of this app and the page says so.
  - The derivation and measurement of alpha_turn ~ eps^(2/(beta+1)) and
    Omega_max ~ eps^(-1/(beta+1)) — the point at which the adiabatic assumption behind every
    published exponent fails. We found no source that states this.
  - The measurement that a fitted exponent comes back systematically low, and by how much.
  - Two errata in the cited papers, stated on the page with the arithmetic.

WHAT DIFFERS FROM THE REAL THING

  - The disk is thin and rigid; a real coin has thickness, so its k differs slightly from 1/4.
  - The table is rigid, flat and perfectly smooth. Below a tilt of roughly 1e-3 radians real
    surface roughness dominates and impacts take over; that regime is outside this model.
  - The commercial demonstration toy is spun on a slightly concave base, not a plane.
  - The centre of mass is held over one point (b = 0), the case the literature analyses.
  - No slipping, no elasticity, no acoustics. The tone is synthesised from the precession rate
    and transposed upward so that it is audible.
  - Integration stops at lift-off, at a tilt of 1e-4 radians, or if the tilt leaves (0, pi/2).

LICENCE

The code in this bundle is released under the MIT licence; see LICENSE.txt. The cited papers
belong to their authors and publishers and are referenced, not reproduced.