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The Shape Sphere

There is a saying in celestial mechanics that ``no coordinate system is ever good enough.'' Indeed, choosing a convenient system of coordinates for the $N$-body problem is one of the greatest difficulties in understanding its dynamics. Here is a brief explanation of a coordinate system for the 3-body problem.

The naïve way to define coordinates for the 3-body problem involves keeping track of three vectors of three components each. In such a case, describing the state of our system at any given time requires nine real numbers:

\begin{displaymath}({\bf x}_1, {\bf x}_2, {\bf x}_3) \in ({\bf R}^3)^3 = {\bf R}^9.\end{displaymath}

By boiling down our system to its essential features, we come up with a vastly simpler way of keeping track of the configuration of our bodies at any given point, using only two real numbers.


Matthew Salomone 2003-07-24