# Determining "boundedness" of particles in an n body simulation

1. Sep 11, 2014

### DivergentSpectrum

is there a numerical method to determine whether two bodies will stay bounded forever in an n body simulation? i know if the energy of a particle orbitting the origin is negative, then it is bounded, where -∫(force)dr+(dr/dt)2/2=energy.
but im curious about a genereral case, where there are more than 2 bodys, and even forces that are not inverse square proportional where i dont necessarily have a potential to subtract kinetic energy from, just the ability to numerically do the line integral of force
∫ [Fx,fy,fz]*[dx/dt,dy/dt,dz/dt] dt
thanks!

2. Sep 11, 2014

### Staff: Mentor

As soon as you have more than two objects, that is not possible in general any more. Energy doesn't help - two objects can move to a closer orbit, giving energy to a third (escaping) object, so a total negative energy does not mean the objects have to stay together (but two of them will).

You can simulate the system for a very long time, of course, but then numerical errors are problematic.

For two objects: if you can write down a potential, then you can check if energy conservation allows a separation. Note that this is not sufficient - a 1/r^3-law with two objects for example has circular orbits, but also unstable trajectories leading either to a separation or a collision.