Hamiltonian for a Dissipative System/ Liouville's Theorem

[kaiserwilhelm]

Homework Statement

Given is the Hamiltonian for a particle in free fall:

H(z,p) = P^2/(2m) + mgz

At time t=0 there is an region given by the constrains:

p1 less than or equal to p less than or equal to p2
E1 less than or equal to E less than or equal to E2

What is the area of the region?

This area moves in the phase space. What are the bounds at t > 0? What is the region's area at this later time?

Suppose we expand the model to include air friction of the Form F= -b(zdot), where b is a constant > 0 and zdot is the velocity. How can the Hamiltonian equations be expanded to include the air friction? Does Liousville's Theorem still apply?

Homework Equations

The area is given by integrating between the parabolas in the phase space:

(E2-E1)(p2-p1)/(mg)

The bounds for t > 0 are given by

(p1-mgt) less than or equal to p less than or equal to (p2-mgt)
E1 less than or equal to E less than or equal to E2 (unchanged)

Substitution into the above area formula yields an conserved area, as we'd expect from Liouville's Theorem.

It's the jump to the Hamiltonian for the dissipative Hamiltonian function that's got me rather stuck at the moment. I know that the phase-space trajectories will converge for the dissipative system, i.e. that the area is no longer conserved and that Liouville's Theorem no longer applies, but I'm unsure of how to show this analytically.

Any advice would be appreciated.

 

[chrisk]

Try writing the Lagrange equation with the friction term included then transform to the Hamiltonian expression. Then ponder, "is this a conservative force and how does this effect Louisville's Theorem?"