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Rotation Invariance

We would also like our plane to be rotation-invariant: that is, we consider any two configurations in the plane to be equivalent if one can be rigidly rotated into the other. This allows us to reduce our configuration description to two pieces of data: the similarity class of the triangle formed by the three masses, and the area of that triangle (which is proportional to the moment of inertia of the system). This is known as taking a quotient of our state space with $SO_2({\bf R})$, the group of rigid rotations of the plane. (In fixing the origin, we have in total taken a quotient by the full Euclidean group of proper rigid motions, $E_2({\bf R})$.)

To accomplish this, we define a system of coordinates on ${\cal X}$ that is invariant under the action of $SO_2({\bf R})$. These are so-called Jacobi coordinates:

\begin{eqnarray*}
\frac{1}{\sqrt{\mu_1}}w_1 &=& \frac{1}{2}(\vert z_1\vert^2 - \...
...^2) \\
\frac{1}{\sqrt{\mu_2}}(w_2 + iw_3) &=& \overline{z_1}z_2 \end{eqnarray*}



where

\begin{displaymath}\frac{1}{\mu_1} = \frac{1}{m_1} + \frac{1}{m_2} \; \; ; \; \; \frac{1}{\mu_2} = \frac{1}{m_3} + \frac{1}{m_1+m_2} \end{displaymath}

are so-called ``reduced masses.''

This is good for two reasons: not only does it accomplish our goal of constructing a rotationally-invariant coordinate system, it also reduces our configuration space from ${\bf C}^3 = {\bf R}^6$ to ${\bf R}^3$.


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