Definitions

Swinging Atwood's machine

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. The Hamiltonian for SAM is:
$H\left(r,theta\right) = T + V = frac\left\{1\right\}\left\{2\right\} M \left\{dot r\right\}^2 + frac\left\{1\right\}\left\{2\right\} m \left(\left\{dot r\right\}^2 + r^2 dot theta^2\right) + g r \left(M - m cos \left(theta\right)\right),$

where g is the acceleration due to gravity and T and V are the kinetic and potential energies respectively.

SAM has two degrees of freedom (as defined by engineers) - r and θ, and a four dimensional phase space defined by, r, θ and momentum variables related to their first derivatives. Energy conservation constrains the motion to a three dimensional subspace in this four dimensional phase space. Additional constants of motion can further constrain the system.

Hamiltonian systems can be classified as integrable and nonintegrable. SAM is integrable when the mass ratio, M/m = 3. An additional non-trivial constant of motion exists for this parameter value. For many other values of the mass ratio (and initial conditions) SAM displays chaotic motion. Research on SAM started as part of a senior thesis at Reed College directed by David J. Griffiths in 1982.

References

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• Tufillaro, N.B. (1982) Smiles and Teardrops, Senior Thesis, Reed College Physics.
• Tufillaro, N.B. (1985) "Motions of a swinging Atwood's machine," Journal de Physique Vol.46, p1495.
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• Yehia, H.M., (2006) "On the integrability of the motion of a heavy particle on a tilted cone and the swinging Atwood machine", Mechanics Research Communications Vol. 33 (5), p711–716.