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

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

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

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.

Less Trivial Example A1 Phase space area preserved again A2

Motion in Phase Space

Orbits Don’t Cross Surface

General Proof Comparing volume and volume a short time later

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

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