Year: 2026
Type: article
Abstract: 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.
Source: European Journal of Physics
Author: Nicholas Tufillaro
Language: English
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: bronze
OpenAlex ID: W7224781693