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).
of Symplectic Advancement
We begin by considering Hamilton's equations as a vector field in phase space.
Exact Time Evolution
Second Order Approximation/Separation of Hamiltonian
Preservation of DpDq
Higher Order Steps
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
Coordinates with Rotation
This system introduces the sinq term into the potential.
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.
Results for the Swining Atwood
More on the Swinging Atwood Machine
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