In a plane, I would like to learn how to compute the initial velocities of 4 bodies (all having same mass), starting off in a square (whose sides are S) so that in absence of all external forces, these 4 bodies rotate in the same circle and whose 4 points always maintain a square. So, only internal gravitational forces on the 4 bodies apply. (I suspect that a small variance from the perfect numbers will cause problems, but let's keep this theoretically Newtonian.)

Note: If the math/physics is easier for 3-bodies (in an equilateral triangle) is easier to use than 4 bodies, then I'd be happy to start with that and not be concerned in this question with 4-bodies. If the problem solutions are sufficiently different and help is available on both, then I'll break one of these cases into a separate question.

I'm guessing that the initial velocity should be tangential to the circle that includes the 4 bodies, and that by symmetry, the speeds should be identical. But, even if right, how do I prove that? And how do I come up with the initial speed?

If the speed is less than the required circle speed, then I am guessing that they spiral into each other gaining speed (but total angular momentum should remain constant), and then they may spiral out. (If the initial speed is 0, then they won't spiral out, but just fall into each other.)

I'm guessing the solution to the circle path is the balancing act where the minimum distance to the origin = maximum distance to the origin = initial radius of the circle. So, maybe I need to figure out the min/max relationship to the initial speed, and then set min=max=initial radius to determine the speed.

I found an article on

3-body spiral that used complex numbers and some kind of Hermite transformation that I wasn't familiar with. Now, I did find this link on

motion and complex_numbers (and I pretty much understand the material that preceeds this section), but seeing the Hermite stopped me from proceeding. An explanation or link of simpler math/physics to help me setup initial conditions would be very nice. If Hermite is absolutely required for me to proceed, the let me know how to learn all about it. Preferably, other more basic concepts can be used.

I'd like to keep the coordinate system fixed at the origin, if at all possible.

If this topic gets too complex, then I'll break it into multiple questions and/or increase points.

You seem to have a pretty good grip on the problem. Advanced math is not needed.

Start by drawing a good diagram. Let r be the radius of the circle on which the masses will travel.

The key physics principle is that the masses will move in a circle IF there is a centripetal force applied. That force is the force of gravity. (F)

F = (m * v^2)/r Gravitation force is F = G mM/r^2

V is perpendicular to radius because you do not want to change that radius

You are correct to appeal to symmetry to say that you can tread each mass the same.

Now get the gravitational force

You have three masses acting on the mass you are discussing.

One, the farthest away and two with equal radial components (The non-radial components cancel out.

F = G (m^2)/(2*r)^2 + 2*G*(m^2/S^2)*cos 45 which = (m*v^2)/r

You know everything but v so calculate

More help if needed.