A Swinging Atwood's Machine (SAM for short) is a
simple Atwood's Machine in which, however, one
of the masses is allowed to swing in a plane
(References).
SAM is a great pedagogical example for learning
about integrable and nonintegrable (chaotic)
behavior in Hamiltionian systems.
A schematic of SAM and the equations describing
its motion (assuming no energy looses) are:
SAM can swing in lots of different ways, showing
motions that are periodic, quasiperiodic, and
chaotic:
SAM also shows some interesting singular
ejection/collision orbits:
As well as an infinite number of distinct
periodic orbits:
SAM is a two-degree of freedom Hamiltonian
system. Because of conservation of energy we
can study the global dynamics of SAM by
examining the Poincare Map for different
values of the mass ration (MU = M/m) of the
nonswinging (M) to swinging (m) mass.
Somewhat remarkably we can show (by explictly exhibiting an additional first integral) that SAM is integrable for MU = 3, but nonintegrable for all other mass ratios except possibly MU = 1 (open problem).