The Swinging Atwood's Machine (SAM) is a compound mechanism comprising a pulley
 and a pendulum and linking two point masses, one of them allowed to swing in a
 plane. This coupled oscillator, basically a variation of the well-known Atwood Machine
 introduced in the late eighteenth century, exhibits an astonishingly complex
 behaviour despite its simple physical description.

We prove the non--integrability of the Hamiltonian system modeling SAM, both in the
 general case and in the case of neglected pulley masses, using basic results from the
 incipient framework of Ziglin-Morales-Ramis theory. The subsequent recollection of
 basics in Analytical Mechanics, Formal Calculus and Differential Algebra, as well as a
 number of alternative or partial proofs of the same basic result, will be assembled into
 a comprehensive survey of algorithms (couched on either symbolic or numerical
 calculations) aimed at detecting chaotic behaviour in general potential systems,
 especially monodromy matrices of the system's linearized higher variational equations.
 Hence, this owes as much to Ziglin-Morales-Ramis theory as it owes to its recent
 extension by Morales-Ruiz, Ramis and Sim—. 

This is a joint work with Juan J. Morales-Ruiz, JosŽ-Phillippe PŽrez, Olivier Pujol, 
Jean-Pierre Ramis, Carles Sim— and Jacques-Arthur Weil