ξ.

Philip T. Wales

2013

The ξ. (pronounced "spring dot") body problem is a unique, two dimensional, three body problem where three point masses are gravitationally attracted to each other. The uniqueness of the system is a linear elastic force between two of the point masses as though they were connected by an ideal spring. The two masses connected by the spring are denoted as the "ξ" subsystem, with the third free mass denoted as the ".".

The equations of motion for the system were derived in multiple cartesian and polar form using both Newtonian analysis and the Euler-Lagrange equations. Code in C++ was developed to numerically solve the equations using primarily the classical Runge-Kutta method. The code was used to create a program that would perform parameter space sweeps which was used to confirm analytic solutions for certain circular orbits derived for the system. Such circular orbits where when the spring was not stretched and the masses were either formed a line or an equilateral triangle.

The ξ subsystem alone proved to be complicated. The true rest length of the system as well as circular orbits for the system were derived. A general solution for the ξ subsystem could not be obtained but the system was greatly reduced to a dimensionless form and independent time.