Swinging Atwood Machine

The swinging Atwood machine (Tufillaro, Abbott & Griffiths, 1984) is one of the simplest mechanical systems exhibiting both integrable and chaotic motion as a single dimensionless parameter — the mass ratio \(\mu = M/m\) — is varied. A heavy mass \(M\) is constrained to slide vertically, and is connected by an inextensible string of total length \(L\) over an idealized pulley to a smaller bob \(m\) that is free to swing in the plane.

Generalized Coordinates

Take \(r\) = string length on the swinging side and \(\theta\) = angle from the downward vertical. The counterweight hangs at height \(L-r\) below the pulley.

• Bob position: \((x,y) = (r\sin\theta,\ -r\cos\theta)\)
• Counterweight position: \(y_M = -(L-r)\); speed \(|\dot r|\)

Lagrangian

Equations of Motion

Applying \(\frac{d}{dt}\frac{\partial\mathcal{L}}{\partial\dot q} - \frac{\partial\mathcal{L}}{\partial q} = 0\) for \(q\in\{r,\theta\}\):

The first term in \(\ddot r\) is the centrifugal pull lengthening the swinging string; the second is the net gravitational force on the string. The \(\ddot\theta\) equation has the familiar pendulum restoring term plus a \(2\dot r\dot\theta/r\) Coriolis-like term coupling radial and angular motion.

Conserved Energy

The simulator displays the relative energy drift \(|E(t)-E(0)|/|E(0)|\) — a sensitive diagnostic of integrator quality. The symplectic Gauss–Legendre integrator keeps it bounded over arbitrarily long runs.

Integrability at \(M/m = 3\)

Tufillaro showed (1985, 1988) that for the special mass ratio \(\mu = M/m = 3\) the system admits a second independent constant of motion and is fully integrable: every bounded trajectory closes on itself, producing a beautiful family of cusped orbits. For all other rational ratios the motion is generally chaotic, with the bob tracing dense space-filling paths. Try the presets to compare.

Numerical Method

The four-state ODE \((r,\dot r,\theta,\dot\theta)\) is integrated with the 2-stage Gauss–Legendre Runge–Kutta method (GLRK4), an implicit 4th-order symplectic integrator. Each step is solved by fixed-point iteration to \(10^{-12}\). For Hamiltonian systems, symplecticity guarantees bounded energy drift over long simulations — essential for distinguishing genuine quasi-periodic motion from numerical artifacts.

Notes on Bounded Motion

• For the bob to swing rather than be yanked up to the pulley, on average we need \(M > m\cos\theta\); a useful rule is \(M \gtrsim m\).
• If \(r \to 0\) the pendulum's angular acceleration diverges (singularity at the pulley). The simulator stops and warns if \(r\) leaves the safe interval \((0.05,\ L-0.05)\).