The Fermi-Pasta-Ulam Simulation
Enrico Fermi, John Pasta and Stanislav Ulam simulated in 1953-55 the oscillation of a chain of springs with nonlinear forces (as a model for a crystal with nonlinear binding forces). They used 16, 32 or 64 masses with equal springs and the MANIAC computer at the Los Alamos Laboratories. They had to solve the following differential equations:
d2xi/dt2 = (xi+1 + xi-1 -2xi) + α [(xi+1-xi)2 - (xi-xi-1)2]
d2xi/dt2 = (xi+1 + xi-1 -2xi) + β [(xi+1-xi)3 - (xi-xi-1)3]
They started the simulation in an eigenmode (sinusoidal pattern of displacements) and observed the changes in frequency-space. The FPU-simulation started the branch of computational physics. Search the internet for details.
This applet displays the motion in real space. The Euler-Cromer method is used for numerical integration. Try for example α=8 and watch for a while until you see the recurrence of the eigenmode.
And here is the source-code: FermiPastaUlam.java
Created with BlueJ 3.0.5 (Java version 1.6.0_26) running on Mac OS X 10.6.8
October 10th, 2011, Martin Lieberherr