We began the final section of the course, on rigid body motion.
Started discussion of multiparticle systems. Recalled for two-body systems how the kinetic energy separated out into two non-interacting contributions: one from center of mass motion and one from the particles' relative motion. Given a separable potential (such as a central force potential, which depends only on the magnitude of the relative separation), the entire two-body problem separates into 2 independent differential equations:
1) describing center of mass motion (force-free, for a central force)
2) giving an orbit equation for the separation 
, due to
interactions between the two particles.
We now attempt a similar separation for n-particle systems (where 
). We start with n vector equations of motion, given by
Newton II for each particle. Summing these equations over all
particles in the system, we found that the sum of interparticle forces
vanished, since Newton III told us that these interparticle forces
come in equal and opposite pairs. We generalized our definition of the
center of mass R, as an average particle position in our multiparticle
system, where each particle's position is weighted by its mass. This
told us that the center of mass position obeyed the same equation as
that of a point particle, located at the center of mass, with mass
equal to the total system mass, acted on by a force equal to the total
external force applied to the system.
We also found that, writing each particle position as a sum of a common center of mass position and an individual relative coordinate, we could separate the kinetic energy into independent contributions from the center of mass motion and the particle motions relative to the center of mass. We discussed why an object's center of mass is also its center of momentum: i.e. why center of mass motion carries all the system's momentum and the momentum contributions of particles' relative motions cancel.
We then considered a second linear combination of our n Newton's equations. To consider rotational motion, we took a linear combination giving the system's total torque relative to the origin on the left hand (force) side. On the right hand side, then, we had the time derivative of the system's angular momentum about the origin. We showed that this total angular momentum about the origin broke up into two independent parts: one due to rotation of the center of mass about the origin, and one due to rotation of the system about its center of mass. Claimed that under a "strong" version of Newton III -- under which interparticle forces are not only equal and opposite, but also point along the line connecting the two particles -- the internal torques cancel. Thus we get a second summed equation, saying that the total external torque applied to the system (relative to the origin) gives the time derivative of the total angular momentum (with its two independent contributions from cm motion about the origin and relative motion about the cm).
At this point, we have taken a system with 3n degrees of freedom and obtained simple equations for 6 of those d.f., having to do with translation of the cm and rotations about it. This is slow progress, and likely to only get messier, so we look for a multiparticle system with fewer degrees of freedom. An obvious way to reduce the degrees of freedom is to consider constrained systems -- and we next discuss a common constrained system with exactly 6 degrees of freedom, which we can hope to solve exactly with our 2 vector equations.
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KB