It seems that popular textbooks on electrodynamics do not discuss how to solve the Laplace equation in ellipsoidal coordinates. I could not find any reference, but there must be references about this. Could anyone give a reference?
As an example, the question can be how to calculate the charge distribution on an ellipsoid by solving the Laplace equation. (I know there is a way to solve this particular problem without solving the Laplace equation, but I want to know how the ellipsoidal coordinates works.)
Best Answer
David Peters (Washington University in St. Louis) has guided development of the Dynamic Wake model (for aerodynamics) for helicopter rotors. This is an application of the solution to Laplace's equation in ellipsoidal coordinates. His former student Dale Pitt was his first student to discover early work by Kinner and Mangler/Squire on aerodynamics for circular wings. Chengjian He is probably the most well known for expanding the dynamic wake model. In ellipsoidal coordinates, the ellipsoid collapses to a circular disk when the eta coordinate goes to zero. The coordinate system includes a natural discontinuity between the upper and lower sides of the disk, which lends itself to describing the difference in pressure on the upper and lower surfaces of a circular wing (or rotating rotor blades). Chengjian He and other later students developed closed form solutions to integrals with the Legendre functions that they could not find in other math books. The dynamic wake model niche may be one of few modern applications of the Laplace's equation in ellipsoidal coordinates.