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.