Swinging Atwood Machine
Ben Haley     Professor Mark Haugan
Picture

Hamiltonian

System of Interest

The Swinging Atwood Machine is an interesting system because it's behavior may be completely studied using advanced classical mechanics and because elements of chaos may be seen in the behavior of the system.  The Swinging Atwood Machine has been
studied in great detail by Tufillaro (others) (see Reference X).  The referenced study
focused on perturbations to the mass ratio, m, at a given tilt angle (q = p/2).  In our
study, we focused on perturbations to the tilt angle at a mass ratio (m = 3) known to produce integrable behavior when q = p/2.

Development of Symplectic Integrators
In order to study the behavior of the Swinging Atwood Machine over time, much
effort was focused on the choice of numerical method.  Traditional favorites, such as
the Runge-Kutta algorithm, which translates Lagrane's equations of motion into a
stepping algoritm, do not conserve energy well over time.  We chose to use symplectic
integrators because of their energy conservation qualities.  In order to verify that
symplectic integrators actually do conserve energy well over time, we simulated
several simple systems.  As we considered increasingly complex systems, the Hamiltonians under consideration became more like that of the Swinging Atwood
Machine (see Swinging Atwood Machine Hamiltonian).

We begin by considering Hamilton's equations as a vector field in phase space.
Exponential Map
Exact Time Evolution
Second Order Approximation/Separation of Hamiltonian
Preservation of DpDq
Higher Order Steps

Application of Integrators to Simple Systems
We now consider the application of the preceeding process to some simple systems.
In all cases, the results predicted by a fourth order symplectic integrator are compared
to those of a fourth order Runge-Kutta algorithm.

Simple Pendulum
The simple pendulum is perhaps the simplest system in physics.  The idealized system considered here has no friction or damping parameters and is not driven.
The Runge-Kutta and symplectic algorithms give the same results when considering the displacement angle, q, as a function of time.  The phase space results of both integrators are the same, also.  However, when the energy of the simple pendulum is calculated using the Runge-Kutta method, physically unreal results are obtained.

Two Dimensional Harmonic Oscillator in Anisotropic Potential

Cartesian Coordinates with Rotation
This system introduces the sinq term into the potential.

Polar Coordinates
This change of coordinate introduces two distinct terms into the kinetic energy.

We are now ready to consider the dynamics of the Swinging Atwood Machine.

Poincare Sections of Phase Space
The dynamics of the system are very neatly captured in phase space "snapshots".
However, the phase space of the Swining Atwood Machine is four dimensional, so we must consider how to "take pictures" of motion through four dimensions.

Henon's Trick

Results for the Swining Atwood Machine

Conclusions