#format jsmath
= Hill's Equations =

There's a name for objects in "toroidal orbits" - Hill's equations, in the "Hill's frame."

This paper:
 . Burns, R., !McLaughlin, C., Leitner, J., & Martin, M. (2000). !TechSat 21: formation design, control, and simulation. In Aerospace conference proceedings, 2000 IEEE (Vol. 7, pp. 19-25).
 . http://formation-control.googlecode.com/svn/papers/00879271[1].pdf

shows the three equations below, and claims they are found here:

 . Hill, George William. "Researches in the lunar theory." American journal of Mathematics 1, no. 1 (1878): 5-26.
 . http://www.jstor.org/stable/2369430?seq=1#page_scan_tab_contents

but I don't see them in that three equation form.  I suspect these are versions from a more recent paper.


$ \ddot x - 2 \omega \dot y - 3 \omega^2 x ~=~ f_x ~~~~ $ x is the radial direction

$ \ddot y + 2 \omega \dot x ~=~ f_y ~~~~~~~~~~~~~~~     $ y is the orbital direction

$ \ddot z + \omega^2 z ~=~ f_z ~~~~~~~~~~~~~~~~         $ z is perpendicular to the orbital plane.  Right handed triad mentioned in Burns, the equation works with left triad, too.