First, we returned to the Hamiltonian. We considered infinitesimal
variations dH, and showed that 
(with 
held constant). Thus the Hamiltonian depends only on
the independent coordinates 
and their conjugate momenta 
. (This
transformation from 
to 
is an example of
a Legendre transformation, which you will frequently run into in
thermodynamics). We then showed that imposing the E-L equation on our
expression for dH led to Hamilton's equations: 2 first order d.e.'s
for each degree of freedom which reduce to the second order E-L
equation when substituted into each other. We then introduced the
notion of phase space, and particle motion as defining trajectories
through phase space, showing that a simple harmonic oscillator
traverses an ellipse through phase space. There are many theorems
about allowed motions through phase space, and it is often by
considering trajectories through phase space that we see the onset of
chaos.
This concluded the formal section of the course. We then began to study applications of mechanics, starting with 2-body central force motion.
We study the motion of 2 particles in 3 dimensions, which interact via
a "central force" -- that is, a conservative force which depends only
on the distance between the two particles, and not on their
orientations or absolute locations. Such a force has a potential V
which depends only on the distance 
. Examples
of central forces include gravity, the Coulomb force, spring forces
between particles, etc.
The system initially has 6 degrees of freedom -- the 3-dimensional
positions 
. We simplify things by separating out
the motion of the center of mass. We defined the center of mass
position 
, then rewrote the Lagrangian in terms of
and the relative separation 
. We
found that cross terms in the kinetic energy between 
and 
vanish. Thus -- since the potential, being
central, depends only on 
-- we have a Lagrangian whose
dependences on and decouple. We will begin
solving the decoupled E-L equations next time.
-- KB