Week 10: Multiple particle systems

Systems of two particles with a force between them that points along the direction between them can be replaced by a "reduced" problem, which treats the motion in the center of mass frame and looks exactly like a single particle having the "reduced mass" which is acted on by one part of the pair of equal and opposite internal forces on the two particles. This is a powerful way to treat two-particle systems that are not subject to external forces. The situation is not so simple when systems include many independent particles, but we can make progress by considering the center of mass frame, and treat only some aspects of the internal motion of the system by focusing on conservation principles and averaging over the various potential complex motions that we will treat as random. (This is the approach of statistical mechanics, where we in effect imagine a huge numbers of different versions of the same system, and average over all the different detailed behaviors that we are not interested in distinguishing.) In both cases, the first step in the process is dividing the motion into the motion of the center of mass, and motion with respect to the center of mass.

Motion of the center of mass (COM):

In a system of N particles moving independently in 3-space, we have 3N generalized coordinates to track for a complete solution. By taking 3 of those coordinates to be the center of mass coordinates, we accomplish a division of the kinetic energy of the system into two separate pieces-- the kinetic energy of the center of mass (as though the entire mass M of the system was moving with COM displacement R), and the kinetic energy with respect to the center of mass (the kinetic energy in an observer frame moving with the center of mass). Often, the potential energy function that is responsible for the system dynamics can also be divided up into an external energy that depends only on the COM displacement R, and an internal energy that depends only on the relative positions of the internal components. When this is true (such as if the external force is gravity), the dynamics of the COM are completely decoupled from the internal motions, so we have a 3-dimensional COM problem that can be solved independently of any internal motion, and the internal problem can be solved independently of the COM motion.

So if we have a system in free-fall, or if the system has no external forces on it, the internal motion is independent of the external (COM) motion. Since the internal motion is due to forces in action/reaction pairs, if each of those forces is associated with a potential energy function, the internal kinetic energy will be changed only by changes in the internal potential energy. This is also true if internal chemical or nuclear energy is released in an explosion. Also, the internal motion cannot generate any net torque, so the internal motion conserves angular momentum. Knowing these conserved quantities may help us answer questions about the internal (relative to the COM) motion, and the remaining degrees of freedom of the internal motion are quite complicated and usually too hard to track.

Two-body Collisions:

One type of internal interaction that is not so hard to track is when you have an elastic collision between two particles. This can be due to a glancing collision between rigid spheres, or can be treated as an interaction due to a potential energy between two point particles. Either way, if we focus on the situation when the two particles start out well separated, and end up well separated, if the collision is elastic we will conserve both energy and momentum. We are tracking the motion of the relative displacement of the two particles, and this motion is restricted to the plane spanned by the relative motion vector and the impact parameter (the vector of closest approach in the absence of any interaction).

Such a two-body problem under a central force was solved in chapter 8 (using reduced mass), but here we may wish to leave the scattering angle an unknown (it depends on the impact parameter) and just use the conservation of energy and momentum to constrain the possible outcomes of the collision. This is best done in the COM frame of the two particles, and then one can transform to any other frame, such as a frame where one particle is initially not moving (often called the lab frame). One quickly encounters many equations relevant to this encounter, and gets rapidly into details of "how to" solve various specific situations, details that are not so interesting. So we will mostly focus on using the general conservation laws (which give you results such as the elastic scattering angle for equal mass particles in the lab frame is a right angle), and we will not talk about integrated or differential cross sections. Such details can be included in a course that is more dedicated to scattering physics, and are covered in the book, but we will merely select crucial topics from Chapters 8 and 14 that relate to important concepts for treating multiple-particle systems.