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 q'_j = q_j + \epsilon (\delta q_j), which
changes the Lagrangian by L' = L + \epsilon dF/dt, the charge Q =
(\sum_j p_j (\delta q_j) ) - F 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
\epsilon about the rotation axis \hat{n} leads to d\vec{x} = \epsilon
\hat{n} \cross \vec{x}. We noted why the Lagrangian is unchanged by
this transformation, to first order in \epsilon. We then plugged
\delta x, F, and p into Noether's theorem to find that the associated
conserved quantity for rotation about the axis $\hat{n}$ is the
\hat{n}-component of angular momentum.

Finally, we considered the example of time translation. Expressed as a
coordinate transformation, time translation causes q_k to go to q_k
+ \epsilon \dot{q}_k. We found the Lagrangian changes by a total
derivative, \epsilon dL/dt, 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 dH/d\dot{q}_i =0. Thus the Hamiltonian depends only on the
coordinates q_i and the conjugate momenta p_i. (This transformation
from L(q_i, \dot{q}_i) to H(p_i, q_i) 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. 


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KB