Unformatted text preview: i + ∂ ˙ p i ∂p i ! + n X i =1 ∂ρ ∂q i ˙ q i + ∂ρ ∂p i ˙ p i ! . Using Hamilton’s equations we have ∂ ˙ q i ∂q i = + ∂ ∂q i ∂H ∂p i and ∂ ˙ o i ∂p i =-∂ ∂p i ∂H ∂q i . Interchanging the derivative these terms cancel one another and we are left with Liouville’s theorem: = ∂ρ ∂t + n X i =1 ∂ρ ∂q i ˙ q i + ∂ρ ∂p i ˙ p i ! = dρ dt . This is the motion of an incompressible ±uid, but in phase space instead of coordinate space....
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This note was uploaded on 12/11/2011 for the course PHY 4241 taught by Professor Berg during the Fall '11 term at FSU.
- Fall '11