Elliptical thoughts

Posted August 16, 2005

For some reason, my MIT card still scans as having borrowing privileges
at the library, even though it really shouldn't unless I start paying them
a (hefty) yearly fee. So I'm exploiting this loophole for the moment
to peruse some nice texts. *Shhh, don't tell them!*

Needham's Visual Complex Analysis is one such tome, and ended up being a pretty nice treatment of the field with a bit of a different twist in presentation: very geometric as opposed to a pure mathematical approach.

The neatest bit was a small section about applications of complex analysis
to orbit problems in physics. You probably know that planets move on
elliptical orbits with the Sun at one focal point: this is a consequence
of gravity being a 1/*r*^{2} force. You probably don't know
that there is another force law that also results in elliptical orbits, a
linear force proportional to *r*, like a big rubber band attached
to the Sun. In that case, the ellipse is centered on the Sun instead of
being at a focal point.

What I didn't know is that there is a straightforward way to transform one situation into the other, by putting the orbit into the complex plane and just squaring the equation... that map converts an ellipse centered on the Sun to one with a focal point there instead, as seen in this cheapo animation. There's a similar transform of the force law, and moreover this generalizes to what are known as "dual force laws" in the Kasner-Arnol'd Theorem, where one exponent in the force law produces the same orbits as a particular other exponent, with the orbit shapes mapping as a power in the complex plane. Very neat, and somehow a result that I never ran across in either my physics or mathematics classes.

Oh, in case you were wondering, hyperbolic orbits with a gravitational force
law correspond to the *repulsive* linear case, like a giant spring
hooked to the Sun.