Simple, stable $n$-body orbits in the plane with some fixed bodies allowed

Question

I'm working on a visual simulator for the $n$-body problem in the plane (here). The goal is to show how complex behavior can arise from the simple inverse-square law of gravity.

To that end, I want to showcase cool stable orbits. I have three of them already:

• moving planet around fixed star
• moving stars orbiting each other
• moving planet on a straight line between two starts

What other simple orbits are there? I'm allowing myself to fix some bodies in place, in order to emulate the real-world situation where one body is much more massive than another.

All answers are appreciated, but I'd especially appreciate pictures and information on how to construct the orbits (initial velocities and positions).

Answer 1

A Lagrange point relative to the positions of two massive objects is a third point with the property that a small object moving under the gravitational influence of the others remains in relative equilibrium. The following image (taken from Wikipedia and licensed under CC) shows the five Lagrange points in a Sun-Earth like system, together with contours of the associated potential:

As the image indicates, there are five Lagrange points labeled $L_1$ through $L_5$. The first three are unstable while the last two are generally stable. Thus, we can see three bodies moving under gravity that maintain their relative positions on a triangle. The size of that triangle can change, but not it's shape. The degree of that stability depends on the relative masses, however.

One specific set of initial conditions in the plane with a gravitational constant of $G=1$ illustrating this type of orbit is given by:

• Body 1:
  • Mass: 4
  • Position: $(0,0)$
  • Velocity: $(0,-1/\sqrt{2})$
• Body 2:
  • Mass: 4
  • Position: $(1,0)$
  • Velocity: $(0,1/\sqrt{2})$
• Body 3:
  • Mass: 0.0001
  • Position: $(1/2,\sqrt{3}/2)$
  • Velocity: $(-\sqrt{3/2},0)$

The evolution of this configuration looks like so:

I think this system would make a great illustration of chaos because the masses have been chosen so that we don't have to deviate very much before things go haywire. If we move the position of the small object just slightly to the left, from $x=0.5$ to $x=0.48$, we obtain the following evolution:

Another very cool point to make in a presentation concerning Lagrange points is that they can be observed in nature. There is, supposedly, a huge amount of space debris located at the Sun/Jupyter $L_4$ and $L_5$ points.

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A second type of fun illustration is simply animation from a random initial configuration. That it, pick $n$ initial masses and place them at $n$ initial locations. One could also pick $n$ initial velocities, though that get's trickier, since the overall picture has non-trivial momentum.

Here, for example, is the evolution of 5 random point masses at 5 random locations:

Answer 2

Although you can witness complex (or chaotic) behavior via a simulation with some of the bodies fixed, it usually pays to run a complete numeric solution to the general problem, since any non-fixed bodies can always run chaotically fairly easily.

If, on the other hand, you wish to fix some of the bodies, you can tweak the general solution by giving them large masses and 0 velocities relative to the smaller ones in the simulation.

The main thing the engine for the solution does is simply calculate discrete updates for (all bodies):

1) Acceleration (from Newton)

2) Velocity

3) Position

Such an engine is best understood by example, so if you have access to Maple, I am including a .pdf with some code on it here.

Even if you don't have Maple, you should be able to pick the code apart and see how it calculates the following 3 animations, which are run by changing N only (N=4, N=3 and N=2).

Chaotic behavior can then be witnessed fairly easily, by fiddling with the initial velocities, which may produce among other things, body crashes and/or escapes.

In the animations below, the initial velocities were tweaked so that the system would be stable, at least for the duration of the animation.