# Playground Swing

> An interactive playground swing with its exact equation of motion, plus the stability
> analysis that decides whether pumping works. Runs entirely in the browser. No accounts,
> no network calls after load, no cost.

## What it is

A canvas swing you pump with the keyboard or on-screen hold buttons, a bell to ring at 60 degrees, and a jump-off for
distance. Underneath it is the full Lagrangian for a rigid rider on a rope whose own body is
a prescribed internal coordinate, integrated with RK4, plus a Floquet stability analysis and
the published Mathieu transition curves.

## The finding it was built around

The usual advice is "you pump a swing by leaning back and forth in time with it." A rider has
two independent things they can do, and they are not the same mechanism:

- Move the centre of mass ALONG the rope (stand up, squat). This changes a coefficient of the
  equation rather than applying a force: a parametric drive. It works only at TWICE the swing
  frequency. At exactly the swing's own frequency it does nothing at all -- substituting
  u = l*theta gives u'' + (g/l - l''/l)u = 0, and for l = l0(1 + eps cos(w0 t)) that bracket
  is identically w0^2. The cancellation is exact in the squat depth, not a small-parameter
  result, and the monodromy trace over a pump period comes out at 2.000000000.
- Turn the body ABOUT its own centre of mass (lean back, lean forward). This is a genuine
  torque at whatever rate you lean, so it works at the swing's own frequency. Its first-order
  amplitude is D = m g Ic L alpha0 / M0 -- exactly proportional to the rider's own moment of
  inertia about their own centre of mass, and identically zero for a point-mass rider.

So the folk sentence is half right, and right for a reason it does not give. The real split is
translation versus rotation, not standing versus seated: both postures use both channels. A
torque adds amplitude linearly and works from a dead stop; a parametric drive multiplies
whatever amplitude is already there and cannot start at all. That is why children start seated
and finish standing, and the app finds the crossing time.

## Verification

449 engine assertions, all passing. Three independent routes to the same numbers: the
nonlinear engine, an exact Floquet analysis of the engine linearised only in theta, and the
published transition-curve series of DLMF 28.6. Tongue boundaries located by bisection agree
with the series to eight significant figures at q = 0.02. The impulsive pump law
(l_long/l_short)^3 per half swing is confirmed both by an angular-momentum oracle and by a
smooth ramp that assumes no impulse at all -- and it disagrees with the one discrete-jump
relation reachable in print, which is reported rather than resolved.

## Sources

Curry 1976; Tea and Falk 1968; Burns 1970; Case and Swanson 1990; Case 1996; Post et al 2007;
NIST DLMF 28.6; CPSC Public Playground Safety Handbook #325. Full citations, and exactly how
much of each was readable, are in CREDITS.txt.

## Pages

- / -- the swing, the lab and the about page, all on one page
- /CREDITS.txt -- citations, what differs from a real swing, what is original
- /LICENSE.txt -- MIT