Collisions of more than two bodies

[greypilgrim]

Hi.

The formulae for the velocities of two bodies after a perfectly elastic or inelastic bodies, let's say in 2D, (e.g. billiard) can be derived from three equations: conservation of energy and conservation of momentum in two dimensions.

But how do you treat collisions of three or more bodies? With each additional body the number of unknowns rises by two (velocity in x and y direction), but the number of constraints is still three.

Is the problem stable with respect to shifting the bodies by arbitrarily small (or virtual) displacements such that there are only two-body collisions to consider?

 

[mfb]

Is the problem stable with respect to shifting the bodies by arbitrarily small (or virtual) displacements such that there are only two-body collisions to consider?

It is not, and in general material properties will be highly relevant for the outcome if the overall process cannot be described as independent two-body collisions separated in time.

 

[greypilgrim]

I once read that the dynamics of Newton's cradle is higly dependent on the shape of the bodies used, using cylinders instead of spheres will not show the same behaviour.

Elsewhere I read that one may treat Newton's cradle as if the spheres are separated by a small distance such that only two-body collisions occur. With this assumptions one can easily derive the observed dynamics by just using the equations for two-body collisions. So is this a coincidence and wouldn't work with cylinders?

 

[mfb]

Apparently not. It would work if the cylinders are actually separated a bit.