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

• Almeida, M.A., Moreira, I.C. and Santos, F.C. (1998) "On the Ziglin-Yoshida analysis for some classes of homogeneous hamiltonian systems", Brazilian Journal of Physics Vol.28 n.4 São Paulo Dec.
• Barrera, Jan Emmanuel (2003) Dynamics of a Double-Swinging Atwood's machine, B.S. Thesis, National Institute of Physics, University of the Philippines.
• Bruhn, B. (1987) "Chaos and order in weakly coupled systems of nonlinear oscillators," Physica Scripta Vol.35(1).
• Casasayas, J, A. Nunes, and N. B. Tufillaro (1990) "Swinging Atwood's machine: integrability and dynamics," Journal de Physique Vol.51, p1693.
• Casasayas, J., N. B. Tufillaro, and A. Nunes (1989) "Infinity manifold of a swinging Atwood's machine," European Journal of Physics Vol.10(10), p173.
• Chowdhury, A. Roy and M. Debnath (1988) "Swinging Atwood Machine. Far- and near-resonance region", International Journal of Theoretical Physics, Vol. 27(11), p1405-1410.
• Griffiths D. J. and T. A. Abbott (1992) "Comment on ""A surprising mechanics demonstration,"" American Journal of Physics Vol.60(10), p951-953.
• Moreira, I.C. and M.A. Almeida (1991) "Noether symmetries and the Swinging Atwood Machine", Journal of Physics II France 1, p711-715.
• Nunes, A., J. Casasayas, and N. B. Tufillaro (1995) "Periodic orbits of the integrable swinging Atwood's machine," American Journal of Physics Vol.63(2), p121-126.
• Ouazzani-T.H., A. and Ouzzani-Jamil, M., (1995) "Bifurcations of Liouville tori of an integrable case of swinging Atwood's machine," Il Nuovo Cimento B Vol. 110 (9).
• Sears, R. (1995) "Comment on "A surprising mechanics demonstration,"" American Journal of Physics, Vol. 63(9), p854-855.
• 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.
• Tufillaro, N.B. (1985) "Collision orbits of a swinging Atwood's machine," Journal de Physique Vol. 46, p2053.
• Tufillaro, N.B. (1986) "Integrable motion of a swinging Atwood's machine," American Journal of Physics Vol.54(2), p142.
• Tufillaro, N.B., T. A. Abbott, and D. J. Griffiths (1984) "Swinging Atwood's Machine," American Journal of Physics Vol.52(10), p895.
• Tufillaro, N.B. (1994) "Teardrop and heart orbits of a swinging Atwoods machine," The American Journal of Physics Vol.62 (3), p231-233.
• Tufillaro, N.B., A. Nunes, and J. Casasayas (1988) "Unbounded orbits of a swinging Atwood's machine," American Journal of Physics Vol.56(12), p1117.
• 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.