An exact solution is presented for a swinging Atwood's machine. This teardrop-heart orbit is constructed using Hamilton-Jacobi theory. The example nicely illustrates the utility of the Hamilton-Jacobi method for finding solutions to nonlinear mechanical systems when more elementary techniques fail.
@article{arxiv.chao-dyn/9302006,
title = {Teardrop and heart orbits of a swinging Atwood's Machine},
author = {Nicholas B. Tufillaro},
journal= {arXiv preprint arXiv:chao-dyn/9302006},
year = {2009}
}