For the Swinging Atwood Machine with any nonzero string mass, meromorphic Liouville integrability is impossible, so the classical integrable case at mass ratio 3 is destroyed.
Anumericalanalysisofchaosinthedoublependulum.ChaosSolitonsFractals, 29(2):417–422, 2006
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Dynamics and non-integrability of the Swinging Atwood Machine with a massive string: chaos, periodic orbits and resonance structures
For the Swinging Atwood Machine with any nonzero string mass, meromorphic Liouville integrability is impossible, so the classical integrable case at mass ratio 3 is destroyed.