# What is the result of a classical collision between THREE point particles at the same precise instant?

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?

• 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? Jan 16, 2013 at 23:10
• Why is this more difficult than 2 points? Jan 16, 2013 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. Jan 16, 2013 at 23:12
• And for 2 particles they are? Jan 16, 2013 at 23:13
• In the n-body problem, collisions of more than 2 simultaneous particles cannot be analitically continuated, see en.wikipedia.org/wiki/…, the "trick" is to disregard them as highly improbable,i.e. the initial data that would produce one has Lebesgue measure zero. Jan 17, 2013 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.

• 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. Jan 17, 2013 at 14:11
• Jan 17, 2013 at 14:17
• 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. Jan 17, 2013 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. Feb 19, 2013 at 12:31
• @Mephisto: Classical mechanics is deterministic when applied to finite-sized objects and fields using continuum mechanics on continua with finite density. The problem is that the notion of point particles is broken, not classical mechanics. Sep 19, 2016 at 17:07

I have not found any reference on this. The following is my approach for point particles, colliding in the same point of space at the same time (elastic collision).

Let us consider 2D motion and the particles traveling towards the collision point. If the masses and momenta were equal, it would be natural to assume by symmetry, scatter back with equal momenta, in the same incoming directions, but in opposite sense.

While if only one particle had different momentum to the other two the symmetry would be along the uneven one. The components of the two equal particles, perpendicular to the symmetry line, should be unchanged (only the sense to opposite). And the components along the symmetry line would be calculated by the balance of the momentum of the incoming particle and the sum of the components of the remaining two (2 times the value of one due to symmetry), which would replicate the case of collision with a particle twice as massive as one of the two equal, whose velocity is reduced by the cosine (because is the component of the original momentum).

As for the case where all momenta and masses are different, I would generalise by balancing each particle with the components of the other two along its direction. This would give three equations, but only two of them would be linearly independent. This would be something similar to the figure (although in such figure the initial momenta are zero and the particles attract each other) Finally note that the mentioned restriction (2D motion), applies also for 3D collisions, because the center of mass motion occurs in one plane.

Indeed, if we choose the lines of motion of two of the particles, these two lines define a plane. The third line will in general be out of this plane. Let us further define the plane containing the three particles in any moment, as the center of mass plane. The projections of the momenta of each of the particles in this plane does not change in time, since the total momenta and their angles with this plane do not. Even more, the components of momenta perpendicular to this plane are equal for all particles, as this plane moves towards the collision point. Hence the name center of mass plane. (if you prefer, I can provide a mathematical proof)