Imagine that. my 59-body solution Is a wee unstable

https://reddit.com/link/1hzfdjk/video/p602ww4iwhce1/player

To improve it, I’d need help with an integral that’s over my head

Working on a solution for an N body system with bodies of equal mass, equally spaced in a circle, orbiting along that circle. I claim there should be a formula for the circular orbital V - given radius, mass and number of bodies.

I failed on repeated attempts to research or derive the formula for the forces acting on each body, and integrate that force across the number of bodies.

So i cheated and solved it numerically - and was stunned how well it worked. 

The cheat: 

• Place the objects in my sim and measure the net force on each body.
• No surprise, a vector toward the center - see the vector view in the video.
• There must be a circular orbit velocity normal to that acceleration, which maintains this distance. 
• calculate the orbital velocity for this acceleration as if it were due to a single mass at the center

so we’re literally measuring the forces on the bodies and working backwards to find an equivalent single mass to orbit - since we already know how to solve that.

Given how well this worked with “manual” calculation i’m inspired to get even more exact. All i need is a formula for that net acceleration vector that I measured in-sim, at the beginning of the cheat.

edit: yes. of course it'll still be unstable.