Liouville's theorem

This animation draws the phase space for the mathematical pendulum
d²φ/dt² + g/L sin φ = 0
or the harmonic oscillator
d²x/dt² + ω² x = 0, (x = φ, ω = (g/L)^1/2).

The unit time is (L/g)^1/2, so that ω = 1.
One can add a friction term proportional to the (angular) velocity.
The simulation computes the evolution of a set of phase points around the red center. The latter's position can be selected with the mouse or by entering its coordinates. One can also choose the number of points, as well as the initial form and the diameter of the set of points.
It is also possible to draw the phase trajectory of the center point.
To get information on the other elements, rest the mouse on the Element to see the corresponding tooltip.

Activities

Check that without friction the phase surface is conserved.
Check that with friction the phase surface is always decreasing.
In the harmonic oscillator all phase trajectories have the same period (they are isochronous). As a consequence, the set of points moves without deformation, only orientation changes.
With the full pendulum, however, the period depends on the amplitude and different points need different times to complete a full revolution: some points lag behind others and the set of points stretches continuously.
(With the pendulum you should check Cylinder to make sure the program keeps the angle between -π and π.)

This is an English translation of the Basque original for a course on mechanics, oscillations and waves.
It requires Java 1.5 or newer and was created by Juan M. Aguirregabiria with Easy Java Simulations (Ejs) by Francisco Esquembre. I thank Wolfgang Christian and Francisco Esquembre for their help.