Another interesting aside is the connection of the differential equation that I am solving to the elliptical orbit discussion on the Azimuth blog -- http://johncarlosbaez.wordpress.com/2015/03/17/planets_in_the_4th_dimension

I am applying a nonlinear term to the second-order differential equation describing a sloshing of a volume of liquid. When this is a single sinusoidal term, it is known as the [Mathieu equation](http://en.wikipedia.org/wiki/Mathieu_function#Mathieu_equation). More generally, when the factor is a set of sinusoidal terms of different frequencies, it is known as the [Hill differential equation](http://en.wikipedia.org/wiki/Hill_differential_equation). I am specifically using Hill because I am trying to approximate the TSI, which is roughly a harmonic combination of 11 year and 22 year periods.

The connection to orbital mechanics comes about because this is the same Hill, [George William](http://en.wikipedia.org/wiki/George_William_Hill) who developed the first comprehensive perturbation theory for determining corrections to the lunar orbital period. The correction essentially comes about because of the non-linear effects of approximating the 3-body problem of the sun-earth-moon system. I believe that the variational corrections to official lunar calendars all have some basis in this physics.

A fascinating account of Hill and others involved in the development of the math is found in the review by Gutzwiller, [Moon-Earth-Sun: The oldest three-body problem](http://sites.apam.columbia.edu/courses/ap1601y/moon-earth-sin%20rmp.70.589.pdf).

> 'The motion of the perigee c0 is obtained to very high accuracy. Poincaré remarks in his preface to Hill’s Collected Papers: ‘‘In this work, one is allowed to perceive the germ of most of the progress that Science has made ever since.’'

Gutzwiller also refers to Hill and Poincaré in his account of the development of [Quantum Chaos](http://www.stealthskater.com/Documents/Math_02.pdf) math. I imagine that this is the basis of Poincaré's quote :)