Classical Mechanics
11/26/2007
Stanford
class with Leonard Susskind.
We study Liouville’s theorem tonight. We’ll prove Liouville’s theorem from Hamilton’s equations. Liouville’s theorem can be thought of as information conservation.
The laws of mechanics are synonymous with a specification for state transition.
A flow through phase space doesn’t condense, it moves as an incompressible fluid. No divergence or convergence. Forbidden.
Not allowed in Classical Mechanics
Ignored degrees of freedom, such as friction can result in multiple paths to the same final state. A point in phase space is while it’s velocity is .
In one dimension incompressible motion implies a constant velocity.
Consider two dimensions.
The density of points must not change, so the number of points inside the rectangle must be constant.
Points per second leaving the four faces must sum to zero.
For a given wall: gives the points per second with velocity crossing a wall with velocity . This can be written as
The quantity
gives the difference in horizontal output vs. input between the left and right walls.
Now do the same for the y-direction.
vanishes for incompressible flow
then divide by
for incompressible flow.
Calculate divergence of phase space velocity field according to Hamilton’s equations.
so that the sum is zero for incompressible fluids!
No conservation of distance, only area.
Example (one dimensions):
, then
Consider coordinate transformations. (Should use x rather than y).
then
Therefore
The product is constant
Plank’s constant represents area in phase space. Canonical transformations, we’ll discuss later. The area of phase space has the units of ??. Area has an invariant meaning.
chaotic systems
coarse graining, means area increases with time.
finite precision in an experiment implies coarse graining.
implies 2nd law of Thermodynamics.
Volume of phase space is a measure of Entropy.
Entropy = log Volume
is the magnetic field.
force on a charge due to magnetic field. is the velocity.
Need a vector potential in order to write a Lagrangian or Hamiltonian.
e.g.
This is the magnetic force equivalent to .
We want to discover the Lagrangian and Hamiltonian
The units for the Action integral are momentum * x or energy * time.
since
Therefore
or
Take this Lagrangian to prove
The canonical momentum is:
or, in index notation,
The portion of the canonical momentum equal to is sometimes called “mechanical” momentum.
or
then
Substitute on the left.
or
The Hamiltonian is
for any Lagrangian. Investigate the Lagrangian:
Need momenta, not velocity, in Hamilton’s equations.
Substitute for
Substitute into the Hamiltonian.
There is no , therefore the magnetic field does no work.
Solve for in momentum
then
Use this expression for the Hamilton’s equations.
is the canonical momentum.
Exercise: verify using Hamilton’s equations.