Electric dance: a Coulombian 3-body problem with strong symmetries

Overview

Three charged particles, two positive (blue) and one negative (red) are released from rest at the vertices of an isosceles triangle (equilateral in the initial setting).
It’s assumed that the particles have the same charge (but for the sign), the same mass (inertia) and that only the electric force acts on them.

The system dynamics will be just driven by Coulombian attractive/repulsive forces.
Anyway, given the strong symmetries in the initial conditions and given the conservation of energy and momentum, the system can be reduced to just a couple of differential equations, since the position/velocity of one of the blue particles is enough to set the positions/velocities of the other two. Here’s a video of the resulting “Electric dance”:

(YouTube link)

In this page there is an interactive simulation where you can play with some different initial condition and some other parameter (but you need to download the free but hugely sized CDF player to run it).

Since Wolfram Demonstration Project as accepted this simulation to be part of their collection of interactive simulations, you can also download the approved Mathematica code and/or the Wolfram CDF file here.

In this simulation the initial position of the rightmost charge is determined just by its angular position ${\theta _0}$ with respect to the origin. Given the fixed initial energy ${U _0}$ of the system and ${\theta _0}$ , the radial position is set by the formula:
${r_0} = \frac{1}{{{U_0}}}\left( { - \frac{2}{{\sqrt {{{\cos }^2}{\theta _0} + 9{{\sin }^2}{\theta _0}} }} + \frac{1}{{2\cos {\theta _0}}}} \right)$ .
In cartesian coordinates the total potential energy is given by
$U\left( {x,y} \right) = - \frac{2}{{\sqrt {{x^2} + 9{y^2}} }} + \frac{1}{{2|x|}}$
In the initial setting with the three charges forming an equilateral triangle inscribed in an unitary circle, and with unitary charges, masses and electric constant $k_0=1$ , it is $U\left( {{P_0}} \right) = - \frac{1}{{\sqrt 3 }}$ (where $P_0$ is the initial position of the rightmost particle).

In this simulation $U_ 0$ is fixed since this factor only changes the scaling of the orbits and not their qualitative shapes. The positions of the other two charges are determined by the position of the first one, since the three must always form an isosceles triangle with the $y$ axis as axis of symmetry and with center of mass in the origin.

The evolution of the system is then governed by the Coulombian forces acting on the rightmost particle due to the presence of the other two particles.

Since the Coulombian force is conservative, the total energy $E$ is conserved: $E=E_0=U_0= - \frac{1}{{\sqrt 3 }}$ (the charges have zero kinetic energy at $t=0$ ). A negative initial energy ensures that the system is bounded and that the charges will never escape to $\infty$ .

The dotted lines shown in the plot are the lines with constant potential $U(x,y)=U_0$ and delimit the region where the blue particles can move. These lines can be touched only with zero speed.

The purpose of this demonstration is to explore the different kinds of trajectories that this system can produce with different starting conditions, that is with different isosceles triangles, by changing the parameter ${\theta _0}$ . Some looks periodic some quasiperiodic. For instance the equilateral triangle setting seems to produce a periodic orbit (with a period of about 39.1s).

References

Steven Strogatz, Nonlinear Dynamics and Chaos, Westview Press, 1994
Coulomb force (wikipedia)
Electric dance (video of this demonstration on YouTube)

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Electric dance

Electric dance: a Coulombian 3-body problem with strong symmetries

Overview

References

Last edited 10 jan 2017

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