PHYS2006: Demonstration Computer Programs

Three-body problem

The three-body gravitational motion simulator is part of a Chaos Demonstrations program and is one of the many demos available in the program.

Accessing the Chaos Demonstrations program. You should be able to access this from the Physics Teaching menu under the title Classical Mechanics. You should be presented with a choice of various demonstrations. They all illustrate chaos, one way or another. I recommend you to look at them all, but for PHYS2006 the one to pick is the Three Body Problem.

Three body problem. The three-body demo shows a particle (or several particles) orbiting under gravity in the field of two fixed planets of masses m_1 and m_2 (the set up is slightly artificial but is done to make the numerical integration faster).

To begin with, set m_1/m_2 to zero and use one particle only. You should see elliptical orbits. Now add more particles (n > 1). The particles all start at the same position, but with a spread of velocities. Note how the orbits never stray very far from each other: a small perturbation in the initial conditions gives a small perturbation in the orbit.

Now set m_1/m_2 non zero, and return to one particle (n = 1). Look at the trajectory to see why we do not try to solve this problem analytically? Use several particles (n > 1). Note how the trajectories diverge dramatically: you can quickly find the particles localised around different planets. The motion in this case is chaotic in general and one of the hallmarks is the extreme sensitivity to initial conditions shown here.

Waves in One-dimensional Crystals

Here are some animations to go with the last part of the course on one dimensional models for crystal lattice vibrations. The animations are generated using Maple worksheets: Maple is a computer algebra package, available on the Physics and Astronomy PCs and on at least some SUCS clusters. If you download the animations to a PC with a Web browser, you will then be able to start up Maple and look at the worksheets. I hope that each one is self-explanatory.

• Chain with a single type of atom: this animation illustrates that any wavevectors k differing by multiples of 2 pi / a give the same physical displacements to the masses. This explains why we can restrict attention to the Brillouin zone.
• Chain with two types of atom: two animations to show the atomic displacements in optical or acoustic modes.

Downloading with Netscape. Netscape may be set up to start Maple automatically when you download the above files. If not, it may prompt you to choose an application to run the files: you can browse to find the Maple executable. Alternatively, it may be easier to have Netscape save the files to disc and then start up Maple in the normal way..