Lecture 4
Today we finished our discussion of symmetries and conservation laws, and introduced the Hamiltonian and Hamilton's equations.
We started by proving one of the most elegant and widely valid theorems in
physics, Noether's theorem. It says that for every continuous symmetry
a system has, there exists a conserved charge Q. Specifically, for a
symmetry transformation 
, which
changes the Lagrangian by 
, the charge 
is conserved.
We studied specific examples of the use of Noether's
theorem. We showed how the conserved quantity associated with
spatial translation is linear momentum. We then considered a system
with rotational invariance (for example, the motion of a particle
under the gravitational influence of a very massive body at the
origin). We reviewed why an infinitesimal rotation by the angle
about the rotation axis 
leads to 
. We noted why the Lagrangian is unchanged by
this transformation, to first order in . We then plugged
and p into Noether's theorem to find that the associated
conserved quantity for rotation about the axis is the
-component of angular momentum.
Finally, we considered the example of time translation. Expressed as a
coordinate transformation, time translation causes 
to go to 
. We found the Lagrangian changes by a total
derivative, 
, only when it has no explicit
time-dependence. We constructed the conserved charge.
Time translation invariance plays a fundamental role in physics: it asserts our belief that the laws of physics governing our world today are the same as those in the past and those in the future. If we had no confidence about this, we would have little motivation to study physical laws whose nature might capriciously change at any time.
The conserved quantity related to time translation invariance thus has deep significance, so much so that it has received its own name: the Hamiltonian H. Physically, it corresponds to the energy, for almost all natural Lagrangians. This is a very physical object, which you have cultivated a strong physical intuition about. It is quite natural to think of the Hamiltonian H, instead of the Lagrangian L, as the primary quantity governing a system's motion, and to phrase physical laws in terms of H.
To derive these laws, we considered infinitesimal variations dH, and
showed that 
. Thus the Hamiltonian depends only on the
coordinates 
and the 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.
--
KB