Lecture 2
Handouts: Survey
Last time, in motivating QFT, we discussed the method of the missing box: our goal is to complete the following diagram of physical theories
Classical Particle Mechanics ------------> Classical Field Theory
| obeys causality, locality
| take continuum limit, many particles
|
|
| quantum effects
| use Schrodinger eq + operators for observables
|
|
V
Quantum Particle Mechanics
That is, we seek to both relativize AND quantize classical mechanics. To do that, we should examine each box (theory) carefully, as well as the arrows (generalization steps).
Today, we discussed the first box: classical mechanics. Reviewed
Newton's second law (NII) as a vector equation, and relationship of
force to potential energy for conservative forces. Claimed that the
vector equation NII could be messy in general coordinate systems,
where both the gradient and time derivatives assume a complicated form
(the time derivative, when coordinate axes depend on instantaneous
particle position, as in polar coordinates 
-- because then
the time derivative of a vector gets contributions due to the time
dependence both of the components and of the coordinate
axes). Introduced Lagrangian form as a coordinate-invariant statement
of Newton's laws; that is, a form which performs the change of
variables from Cartesian coordinates automatically.
Defined the Lagrangian L as the kinetic energy T minus the potential
energy V; this is a function of the coordinates 
and their time
derivatives 
. Calculated L in polar coordinates, to see the
effect of time-dependent axes. Claimed Newton II always results in the
equation
called the Euler-Lagrange equation. The equation comes with a caveat:
when we take partial derivatives, we treat and as
independent coordinates: that is, we ignore the relationship between
and its time derivative, and hold one fixed while varying the
other to take the partial derivative. This is a weird rule, hard to
justify, but it makes the algebra much simpler, so we learn to live
with it.
Checked that E-L at least gives us back Newton II for Cartesian
coordinates; then calculated the E-L equations in polar coordinates,
for a rotationally symmetric potential. The E-L equations told us two
things: the r equation revealed that angular motion involves a
centrifugal force 
in the radial direction, so that
to keep r constant (circular motion) we had to provide a compensating
centripetal force. The 
equation told us that angular momentum
is conserved. Deriving these facts directly from Newton II takes a bit
more effort.
-- KB
