The swinging Atwood's machine (SAM) is a mechanism that resembles a simple Atwood's machine except that one of the masses is allowed to swing in a two-dimensional plane, producing a dynamical system that is chaotic for some system parameters and initial conditions. Specifically, it comprises two masses (the pendulum, mass m and counterweight, mass M) connected by an inextensible, massless string suspended on two frictionless pulleys of zero radius such that the pendulum can swing freely around its pulley without colliding with the counterweight. The conventional Atwood's machine allows only "runaway" solutions (i.e. either the pendulum or counterweight eventually collides with its pulley), except for M = m {\displaystyle M=m} . However, the swinging Atwood's machine with M > m {\displaystyle M>m} has a large parameter space of conditions that lead to a variety of motions that can be classified as terminating or non-terminating, periodic, quasiperiodic or chaotic, bounded or unbounded, singular or non-singular due to the pendulum's reactive centrifugal force counteracting the counterweight's weight. Research on the SAM started as part of the 1982 senior thesis Smiles and Teardrops (referring to the shape of some trajectories of the system) by Nicholas Tufillaro at Reed College, directed by David J. Griffiths.
Equations of motion
The swinging Atwood's machine is a system with two degrees of freedom. One may derive its equations of motion using either Hamiltonian mechanics or Lagrangian mechanics. Let the swinging mass be m {\displaystyle m} and the non-swinging mass be M {\displaystyle M} . The kinetic energy of the system, T {\displaystyle T} , is:
T = 1 2 M v M 2 + 1 2 m v m 2 = 1 2 M r ˙ 2 + 1 2 m ( r ˙ 2 + r 2 θ ˙ 2 ) {\displaystyle {\begin{aligned}T&={\frac {1}{2}}Mv_{M}^{2}+{\frac {1}{2}}mv_{m}^{2}\\&={\frac {1}{2}}M{\dot {r}}^{2}+{\frac {1}{2}}m\left({\dot {r}}^{2}+r^{2}{\dot {\theta }}^{2}\right)\end{aligned}}}
where r {\displaystyle r} is the distance of the swinging mass to its pivot, and θ {\displaystyle \theta } is the angle of the swinging mass relative to pointing straight downwards. The potential energy U {\displaystyle U} is solely due to the acceleration due to gravity:
U = M g r − m g r cos θ {\displaystyle {\begin{aligned}U&=Mgr-mgr\cos {\theta }\end{aligned}}}
We may then write down the Lagrangian, L {\displaystyle {\mathcal {L}}} , and the Hamiltonian, H {\displaystyle {\mathcal {H}}} of the system:
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