Many-Particle Systems - Cinderella Documentation
Description: The user's guide to the Interactive Geometry Software Cinderella

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# Many-Particle Systems

Sometimes it is desirable to study the interaction of many highly interrelated particles. Imagine, for instance, the balls on a billiard table, a collection of electrons, or a system of several bodies in space. In such a case every particle may exert a force on every other particle. We will briefly discuss these three cases here.

### Charged Particles

Suppose we want to study the interaction of 20 electrons. There are thus 190 different pairs of electrons, with a force of interaction for each pair. One could in principle construct such a particle system by creating 20 charged free masses and then inserting 190 Coulomb Forces. However, in CindyLab there is a much easier way if doing this. After the masses have been created and equipped with charges (using the Free Mass inspector), one can simply open the Environment inspector and check the box Charges cause forces. Then all the particles will interact immediately.

Charges cause forces

The picture below shows the situation of several charged particles trapped in a cage of bouncers. CindyScript was used to make fast particles appear blue und slow particles appear red, with the rest of the spectrum for intermediate speeds.

Many electrically charged particles

The corresponding fragment of CindyScript used for this drawing is the following:

    pts=allpoints();
    f(x):=hue(max((min((0.5,x)),0.0)));
    forall(pts,#.color=f(#.kinetic))

### Gravitational n-Body Systems

In complete analogy to systems of charged particles it is also possible to work with systems of particles that attract each other by gravity, such as systems of celestial bodies. For this one simply has to check the corresponding button in the Environment inspector.

Gravity between masses

The following two pictures show two many-body systems subject to gravitational forces. The first one is a very remarkable figure-eight solution to the three-body problem (which actually is a sporadic example that was discovered only recently). The second picture shows a chaotic system of five bodies. For the simulation only the masses together with their initial velocities have to be drawn and the Gravity between masses button to be checked.

### Masses as Balls