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!

           

Liouville’s theorem

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

 

Velocity dependent forces:  magnetic fields

 

*         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.