Elastic collisions between multiple bodies

Well, i am not sure what u mean by the final state of the point masses .
If you mean the final velocities, then u can use these relations :
Newton's law of restitution : U(1) - U(2) = -V(1) + V(2)
and
the consevation of Kinetic Energy : K.E(before) = K.E(after)

However, i am not quite sure about two or more particles, i mean...how does the law of restituion apply here.

 

As i said, i am not quite sure about the law of restitution, i think it may be applicable in this way :
If u are saying that u have 4 variables this implies that u are dealing with Four Particles . Am i right ?

If yes, then my guess is that :

1st eq : conservation of Kinetic energies
2nd eq : newton's law of restitution for particle 1 and 2
3rd eq : newton's law of restitution for particle 1 and 3
4th eq : newton's law of restitution for particle 1 and 4

4 equations, 4 variables.

 

You will have to apply momentum conservation (momentum is a vector, so angles will come into this) and energy conservation to solve this. And you´d better start with two particles.
I did not know that Newton posited a "law of restitution".

 

http://en.wikipedia.org/wiki/Restitution_coefficient
CoR = 1 is the definition of an elastic collision.

 

And the reason there isn't is because the differential equations are not Lipschitz continuous for this non-real problem where point masses don't interact unless they are touching, and how they interact is underspecified. There are other examples where the differential equations are not Lipschitz continuous and hence there isn't a unique solution. Norton's dome, collections of point masses interacting gravitationally that under just the right conditions don't collide but yet send particles off to infinity in finite time, supertasks involving an infinite number of point masses arranged just so, etc.

The places where Newtonian mechanics fails to be deterministic are either non-realistic (point masses that don't interact unless they are touching) or are a space of measure zero.