From: Dan Asimov <dasimov@earthlink.net> To: math-fun <math-fun@mailman.xmission.com> Sent: Sat, December 11, 2010 11:56:54 AM Subject: Re: [math-fun] Newton's cradle I've long assumed that, ignoring quantum effects, at least Newtonian mechanics is a continuous function of its inputs. Including things like steel balls bouncing off each other. (Of course, as RWG points out, a small error like an off-center collision can propagate into a large error. But in this theoretical problem we are assuming no built-in error like this.) Is it really true that for a triple collision in the plane the result depends on which sequence of double collisions is used to calculate it? (Even in the limit as the discrepancy from a perfect 3-way collision approaches 0 ?) --Dan << I've never been convinced that these problems are even well-defined, for the same reason that a simultaneous collision between three bodies in the plane is ill-defined. In that case, it's immediately obvious that you get completely different results according to how you decompose into a sequence of 2-body collisions.
If in d dimensions n bodies emerge from a scattering, there are dn variables, the components of the momenta of the bodies, and d equations, expressing the conservation of total momentum. If the final state energy is known, for example when the scattering is completely elastic, or in the radioactive decay of a particle, there is one more equation. A 2-body final state in 3 dimensions has 6 variables and 4 equations. We need 2 more constraints for a unique solution, for example, the direction of one of the emitted bodies. In the center-of-mass system, in which the total momentum is zero, the kinetic energy of each body is uniquely determined. This is why alpha and gamma radiation spectra are sharp lines. But the scattering angle is not determined without knowing the details of the incident bodies and the dynamics of the scattering process. With a 3-body final state, there are 9 variables, and even when the final state energy is known and in the center-of-mass system, the partition of kinetic energy among the 3 bodies is not determined. Beta decay has a 3-body final state, the daughter nucleus, an electron, and an antineutrino. The electron spectrum is continuous. The final outcome of an individual collision depends on details of the incoming bodies. If the bodies are classical point particles, we need to know their incident trajectories and the force law between them. If the bodies are finite, then their internal structure, moments of inertia, rotation, elastic vibrations, damping, etc., will figure into the final outcome. Even the free rotation of a rigid body is a messy problem. -- Gene