__Hamiltonian____
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__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).__
__

__General Discussion
of Symplectic Advancement__

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

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.

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__

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__Results for the Swining Atwood
Machine__

__Conclusions__

__Addenda__
__References__
__Links__

More on the Swinging Atwood Machine

More on the Department of Physics and Astronomy
at Purdue University

More Undergrad Research

More on Prof. Mark Haugan

More on Ben Haley
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