Today we finished our discussion of symmetries and conservation laws, and introduced the Hamiltonian.
We continued studying specific examples of the use of Noether's
theorem. We first reviewed 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 
, F, and 
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 
. We found the Lagrangian changes by a total
derivative, 
, only when it has no explicit
time-dependence. We constructed the conserved charge and found it
coincided with the Jacobi integral discussed earlier, which is
constant whenever L has no explicit time-dependence. We thus see why
the Jacobi integral is conserved.
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, as we showed in class for T with quadratic and linear terms in the coordinate velocities. 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. We will do this shortly.
At this point, we wrapped up the discussion of symmetries and
conservation laws. We foreshadowed the power of Noether's theorem in
field theory, which deals not with particles at positions
, but with fields (like the electric and magnetic fields)
that vary over space and time: 
. The motion of
these fields is also governed by an action principle, where the action
is an integral over spacetime of a Lagrange density, varying
over space, which depends on 
, 
, and
. Under symmetries, the fields transform such that
the Lagrange density changes only by derivative terms, 
, where 
ranges over all four
spacetime components. Noether's theorem then gives us a conserved
current, 
, whose conservation law
, means, when written out explicitly,
. That is,
we get a local charge conservation law, relating changes in charge
density within a region to inflows and outflows of current from that
region. In this way we get another kind of conservation law --
conservation of electric charge -- as a consequence of symmetries and
Noether's theorem.
--
KB