Why three? A cone-geometry derivation of the integrable swinging Atwood's machine

The swinging Atwood's machine can be used in classical mechanics courses to study chaotic and integrable dynamics. The standard demonstration of integrability proceeds by a canonical transformation at μ = M/m = 3; however, it arrives without motivation. This paper replaces the ansatz with a derivation. Writing the kinetic energy as a Riemannian metric on configuration space identifies the pulley as a cone point of total angle 2π/√(μ+1). At μ = 3 the angle is π, the cone admits a smooth double branched cover, and on the cover the Hamilton–Jacobi equation separates in parabolic coordinates. The cone condition is necessary but not sufficient: the kinetic and potential terms are shown to require μ = 3 independently. The treatment is self-contained, assumes no prior differential geometry, and is intended for advanced undergraduate and beginning graduate courses; nine problems, sorted by classroom setting, are supplied.