Coordinates

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Section 21.6 Coordinates

The time has come to choose a coordinate system. We have argued that the reduced mass central force problem is spherically symmetric in nature. Therefore, it will be to our advantage to use spherical coordinates, defined in the spherical coordinates section of Appendix B.1, rather than the more comfortable Cartesian coordinates \(x\text{,}\) \(y\text{,}\) and \(z\text{.}\)

In fact, in the present classical mechanics context, we can do even better. For a central force:

\begin{equation} F = f(r) \,\rhat\tag{21.6.1} \end{equation}

the force, and hence the acceleration, is in the radial direction. Therefore, the path of the motion (orbit) will be in the plane determined by the position vector \(\rr\) and velocity vector \(\vv\) of the reduced mass at any one moment of time. Since there is never a component of force out of this plane, the subsequent motion must remain in the plane. In this plane, choose plane polar coordinates:

\begin{align} x \amp= r \cos\phi\tag{21.6.2}\\ y \amp= r \sin\phi\tag{21.6.3} \end{align}

Notice that many textbooks choose to call the angle of plane polar coordinates \(\theta\text{.}\) We choose \(\phi\) so that plane polar coordinates can be seen as a cross-section of spherical coordinates through the \(x\text{,}\) \(y\)-plane, i.e. for \(\theta=\frac{\pi}{2}\text{.}\)