# Minimum number of charges in an oscillatory system

I was just curious actually.

Is it possible to release a set of point-sized charges in space, each particle at rest, such that after a period of time, the system returns to its initial configuration of positions. The particles can have unequal masses and charges.

(Newton's laws and Coulumb's law apply, relativity doesn't. Induction of charge does not occur, charges cannot touch each other in a valid example)

It looked easy at first, but I couldn't find any example of the same. Can the statement be disproved, maybe using Gauss law and unstable equilibrium?

P.S. If it is possible, what is the minimum number of charges required to achieve this?

P.S.2 @rpfphysics has noted that the question is trivial if we allow charges to pass through each other. Hence I'm asking for examples where they do not.

• why not two? A positive and negative charge accelerate towards each other, pass through each other, reach a maximum height. Accelerate back and the process repeats. The nature of the oscillation is mathematically very different to a spring though – lucky-guess Jun 10 '17 at 9:02
• scholarpedia.org/article/… You may wish to look at how the n body problem applies to systems in dynamic equilibrium where the initial velocities are zero – lucky-guess Jun 10 '17 at 9:06
• There's no reason, in principle, why they can't exist, but asking for an initial condition at rest sounds rather too restrictive to me. I doubt this has been explored, even for the gravitational case where we do know several periodic solutions (physics.stackexchange.com/questions/83633 for examples). – Emilio Pisanty Jun 10 '17 at 10:07
• @EmilioPisanty If the initial condition is not rest, then the trivial solution is a pair of particles revolving around a centre. – ghosts_in_the_code Jun 10 '17 at 10:08
• Yes. Is the point of this question to continually ramp up the level considered "trivial" to rule out any examples that get pointed out? Finding periodic solutions is doable but requires intensive numerical work, and you're proposing two major modifications of limited appeal to mathematicians, so I wouldn't hold my breath. If it's been explored, I would expect clear links to the choreography results of Carles Simó from the linked question, so I would recommend that as a starting point for your literature search. – Emilio Pisanty Jun 10 '17 at 10:17

Before working on a general condition for this, (which is a harder question), I can show this problem has answers if we involve 3 charges (at least 3 charges, since two charges released at rest will not behave periodic).

So the way I want to show this is not to mathematically drive path equations for charges and show that they behave periodically (though that would be an ideal answer and I'll work on it).

Instead, for now, I used a computer simulation to show that this is possible. Back in my high school days I made a simulation program in which one could play with charges and learn high school electrostatics.

You can download the program here if you want, however note that it's just a student project so you will find a lot of bugs. Plus, it is not English so you might sometimes need to use your sixth sense.

Here's the video simulation of three charges which are released at rest, and they behave periodically so, yes, the problem has answers. Just check out this video: Initial Configuration:

Each particle is 100 grams.

-1 uC @ (0,0)

+1 uC @ (20 cm, 40 cm)

+1 uC @ (-20 cm, 40 cm)

The trace image of the system roughly looks like this: (It is barely notable that this image corresponds to a slightly different config. of charges than the animated image above)

• @ghosts_in_the_code I updated the video & added information – Moctava Farzán Jun 10 '17 at 17:00