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Coordinates on the Shape Sphere

The above change of coordinates is an isometry; hence we calculate that the moment of inertia (also called the ``size of the configuration'') of the system in our space is

\begin{displaymath}I = \vert z_1\vert^2 + \vert z_2\vert^2 = w_1^2 + w_2^2 + w_3^2 = \rho^2, \end{displaymath}

where $\rho$ is the usual spherical coordinate. If we fix the size of our configuration to be constant (we'll choose 1), we get the level surface $\rho = 1$, the unit sphere in ${\bf R}^3$.

The resulting sphere is called the shape sphere. A point on the shape sphere represents an oriented similarity class of triangles: that is, each point on the sphere represents a triangular configuration with the same internal angles and relative arrangements of the bodies. (Swapping two vertices of the triangle, for example, gives an ``opposite'' configuration.)

A sphere is a two-dimensional object; hence the state of our three bodies can now be described by only two quantities: the spherical angles $\theta$ and $\phi$ representing longitude and latitude, respectively, on the shape sphere.

The latitudinal $\phi$ coordinate is related to the signed area of the triangle defined by the bodies. Specifically, $\cos \phi$ is the ratio of the signed area of the triangle to the area of an equilateral triangle of the same moment of inertia.

Those configurations in which the bodies are collinear will have zero signed area and lie long the equator of the shape sphere; the configurations in which the bodies lie at the vertices of an equilateral triangle will sit at the north or south pole of the shape sphere, depending on the orientation of the triangle.

Within the equator there are three configurations which, in the case of equal masses, have one body exactly at the midpoint between the other two. These are called Euler configurations and are labeled $E_1$, $E_2$, and $E_3$ depending on which mass sits at the midpoint. The north and south poles, corresponding to equilateral configurations, are called Lagrange points and are labeled $L_+$ and $L_-$ respectively. These five configurations have the property that if all three bodies start in one of these configurations with zero initial velocity, they will remain in that configuration until all three collide. (This is called homothetic motion, and the associated configurations are called central configurations. Euler and Lagrange showed that these five are the only central configurations for the three-body problem.)

Along the equator there are three excluded points corresponding to (trivially collinear) binary collision configurations. These are labeled $C_1$, $C_2$, and $C_3$ according to which body does not collide, and were excluded a priori from our shape space.

\includegraphics{shapesphere.eps}

The longitudinal $\theta$ coordinate on the shape sphere relates to the distances between the bodies. The Euler configuration $E_1$ sits on the prime meridian $\theta = 0$, with the other Euler configurations at $\theta = 2\pi/3, 4\pi/3$.


next up previous
Next: Symbolic Dynamics and the Up: The Shape Sphere Previous: Rotation Invariance
Matthew Salomone 2003-07-24