During my senior year at Rose-Hulman, I studied V.I Arnold’s Mathematical Methods of Classical Mechanics with my advisor, Dr. Finn. The first two quarters were spend reading through the book, working through exercises in Arnold’s text, and making connections to the two tensor calc classes we had taken from Dr. Finn. It was a rewarding to compare some of the topics Arnold discusses more intuitively with the discussion in Frankel’s The Geometry of Physics, the text we used for the tensor calc classes.
During the third quarter, I studied symplectic integrators as an application of the theory I learned thus far in the year. Our goal was to reproduce the results of Symplectic Maps for the n-body Problem, by Wisdom and Holman. Specifically, they were able to develop a symplectic integrator that allowed them to estimate features of the orbits of Mars through Pluto over one billion years. Note that Pluto was still classified as a “full-up” planet back in 1991.
One of the characteristic features of symplectic integrators is that they produce a bounded error in the Hamiltonian for all time. Even though I’ve read a proof of this (see Hairer’s Geometric Numerical Integration, for example) and I’ve run the computations, I’m still impressed that they work like they do.
Here’s a small video of the orbits over 100K years, considerably sped up, of course. Note that the time step used in the integration is one year.
Update 2 May 2014
In the Spring 2014 I used the symplectic integrator to [estimate the largest Lyapunov exponent of Pluto’s orbit.