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Planar Configuration

Any three points of ${\bf R}^3$ define a unique plane. Thus, at each point in time we will restrict our attention to the 2-dimensional plane that contains the three bodies.

Also, we would like to fix an origin for this plane in a natural way. Physics tells us that the center of mass is the best choice, the point around which all of the system's mass is ``equally distributed'' in a sense. The center of mass is computed by taking the average of all the planets' positions, weighted by their respective masses:

\begin{displaymath}{\bf cm} = \sum_{i=1}^3 m_i{\bf x}_i \end{displaymath}

To make coordinates on our plane convenient, we define the origin of the plane to be the center of mass of the system.

A point in a plane with a fixed origin can be described by a complex number. This means we have reduced our description significantly to six, rather than nine, real numbers, with a linear relation between them effectively reducing it to four complex numbers. We'll call this state space ${\cal X}$:

\begin{displaymath}{\cal X} = \{ (z_1,z_2,z_3) \in {\bf C}^3 \; : \; m_1z_1 + m_2z_2 + m_3z_3 = 0 \} \end{displaymath}

In fixing this origin we are disregarding any of the motion of our system with respect to a ``stationary'' observer.


next up previous
Next: Removing Collisions Up: The Shape Sphere Previous: The Shape Sphere
Matthew Salomone 2003-07-24