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Classical Mechanics is said to be deterministic, a statement that nearly always is followed by that quote from Laplace, something like

If at one time, one knew the positions and velocities of all the particles in the universe, the laws of science should enable us to calculate their positions and velocities at any other time, past or future.

I always scratch my head after hearing/reading that. If 3 or more rigid point particles happen to collide elastically at the same precise instant, is it not impossible to predict the resulting trajectories? If it is possible, how?

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Of course, a two-particle classical collision is easily solved by examining the problem in the center of mass reference frame, where both conservation of energy and momentum together allow to solve the problem... The question is about 3 classical point particles colliding exactly at the same instant. How to solve the problem? And if it cannot be solved, why is it said that classical mechanics is deterministic? – Mephisto Jan 16 '13 at 23:10
Why is this more difficult than 2 points? – Keep these mind Jan 16 '13 at 23:10
@Gugg: because (if I am not wrong) the two conditions (conservation of energy and conservation of momentum) are not enough to determine the resulting system of equations in the case of three or more particles. – Mephisto Jan 16 '13 at 23:12
And for 2 particles they are? – Keep these mind Jan 16 '13 at 23:13
In the n-body problem, collisions of more than 2 simultaneous particles cannot be analitically continuated, see…, the "trick" is to disregard them as highly improbable,i.e. the initial data that would produce one has Lebesgue measure zero. – Jaime Jan 17 '13 at 0:11

Taking the case of point particles and "contact" collisions seriously actually causes trouble even in the two dimensional case: the instantaneous forces are necessarily infinite even if the impulses remain finite.

The solution to that problem--to recognise that all real particles interact via fields over non-zero distances--solves the three particle problem as well. You just integrate the equations of motion (possibly numerically as this may not be easy in closed form).

This isn't necessarly in the 2 particle elastics case because conservation of energy and momentum fully constrain the outcome allowing us to elide this question in a introductory presentation.

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I'm no classical mechanic, but I've read about catastrophic solutions of the classical three point-particle + gravity problem where particles collide. Then the equations of motion become singular and the evolution cannot be uniquely continued. These solutions form a "set of measure zero" in the solution space (I confess I don't know the measure used). I believe this issue is related to the fact that point particles are always singular and need regularisation (finite radius or similar), even when appearing in conjunction with field theories. I'll try to find a ref. Hopefully an expert comes on. – Michael Brown Jan 17 '13 at 14:11
Hmmm.hadn't seen that, but I'm not deeply surprised. The problem is the point particles, of course. At some point, as the distance scales get very short you have to give up the classical realm because quantum field come to dominate the interaction. In any case, the link is fascinating. – dmckee Jan 17 '13 at 15:05
@dmckee, Thanks for your answer. However, my question is about classical mechanics and how can it be said to be 'deterministic' if a simple collision of three particles cannot be solved. How is it that Lagrange and others before the 1930s thought that we could theoretically predict the future and see the past if we were able to know all positions and momentums of the particles in the Universe and so on... All within the framework of classical mechanics. But, again, thanks for attempting an answer. The question remains open, namely, I still don't understand why Classical Mech is deterministic. – Mephisto Feb 19 '13 at 12:31

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