Classical Mechanics
12/20/2007
Stanford
class with Leonard Susskind.
Goal: create a body of amateur Physicists. people are interested in physics, but not employed by a University. For people who don’t otherwise have access to a graduate course.
Poisson brackets formulation were used to describe phase space and flow in phase space.
Imagine a rotation of a system. Coordinates are . The rotation transformation may mix the p’s and q’s.
Antisymmetric
Linear
Product rule
(the Kronecker delta)
and
The momentum is related to the partial derivative.
Proof:
• 1st show that the relation is true when is a polynomial.
Start with a constant term. Any constant will do.
by product rule
Now try 1st power of .
The pattern is that of a derivative. The proof is by mathematical induction.
and similarly
as before,
Example:
then
Question: what are the allowable transformations that preserve the Poisson Bracket properties?
Example of a transformation.
does not preserve properties since ?
However
does preserve the properties.
We need to check only since by antisymmetry!
Therefore the structure is preserved.
Canonical transformations keep the relations unchanged.
Investigate an infinitesimal transformation. (suppress indices)
Drop infinitesimal quadratic terms.
It is necessary and sufficient that the following be true in order to maintain the Poisson Bracket structure:
Assume a generator function . Now substitute into a canonical transformation.
Always defines a canonical transformation.
So is also a generator of another flow, another canonical transformation.
to get
therefore since
then
Consider canonical transformations that do not change the energy. That transformation is a symmetry.
Consider the time derivative of
Don’t know where this goes:
If the energy is not changed, then
Generator commutes with the Hamiltonian so that is conserved.
Example: A canonical transformation (a rotation). Let
Let
Therefore is conserved under the rotation.
Make up a new system. Let
then
Therefore, circular motion.
Example:
then
Homework: Prepare for Quantum Mechanics next semester. Look at iTunes “Quantum Entanglement, the Logic of Discrete Systems”.
The end of the semester.