Mike Henderson
One of the things I like best is to be able to take a mathematical object and visualize it. Other pages at this site deal with the math, this page is just a bunch of pictures that I have made.
Also check out the pages (mine)
- Coupled Pendula
- Clamped Elastica
- Invariant Manifolds in the Lorenz System
- And the pages linked from multifario's page
Click on the images to see a larger version (or in some cases an animation).
These first are from a study of the equilibrium configurations of an elastic rod which is clamped in such a way that the two ends of the rod lie on a common line. A force can be applied (either pushing the ends together or pulling them apart), and two torques (one twisting the rod and the other bending it).
The result is a surface in the parameter space (force and two torques). The surface is complicated, and each point on the surface corresponds to a shape of the rod. The surface does not cross itself, but in this projection onto the parameter space it appears to,
The following images give some idea of the shape of the surface.
The next example is the motions of a pair of pendula on a common axle, coupled with a spring. There are three parameters, the strength of the spring (k, fixed in the images below) and two torques (I1, I2) applied to each pendulum respectively (the horizontal plane in the images below). The vertical axis bellow is the period of the motion.
The first image is the surface of stationary points (where the pendula don't move) and show the separation of the pendula as a function of I1 and I2.
In the image above the spring strength k is fixed. If the surface is projected down onto the (I1,I2) plane and the fold lines are drawn (i.e. where the number of stationary points change), and k is allowed to vary, we obtain the image below.
Finally (for the pendula) is the surface of period as a function of (I1,I2) for fixed k.
And some snapshots of the algorithm used to compute the surface:
