1. Physics 319 Classical Mechanics
G. A. Krafft
Old Dominion University
Jefferson Lab
Lecture 24
G. A. Krafft
Jefferson Lab

2. Liouville: Elementary Argument
Liouville Theorem: Phase space volume is conserved under Hamiltonian motion

3. Falling Body Variables x (directed down), p. Hamiltonian
Equations of motion

4. Phase Space Area Before and After
Phase space area enclosed by a closed curve is preserved for any Hamiltonian phase space flow. This is the statement of the 1 degree of freedom version of Liouville’s Theorem.

5. Less Trivial Example A1
Phase space area preserved again A2

6. Motion in Phase Space

7. Orbits Don’t Cross Surface

8. General Proof
Comparing volume and volume a short time later

9. Divergence Theorem Most common in 3-D space in electromagnetism
Divergence operation
Equally applicable in 6-D phase space. Suppose have 5-D closed hypersurface in phase space

10. Liouville Theorem Rate of change of phase space volume
By Divergence Theorem
By Hamilton’s Equations of Motion