Phase space

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.
It is also possible to draw the phase-trajectories.
To get information on the other elements, put over them the mouse pointer to see the corresponding tooltip.

Activities

Use the default settings to draw the phase space for the mathematical pendulum.
Discuss the physical difference between phase trajectories that go around the origin and those that remain always above (or below) it.
Check that the period depends on the amplitude, so that different points need different times to complete a full revolution: some points lag behind others.
Check Harmonic to draw the phase space for the harmonic oscillator and discuss the differences with the previous case.
Add a bit of friction (say γ = 0.1) and repeat the previous steps.
What happens with a lot of friction (γ > ω = 1) ?
(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.