Unformatted text preview: 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 ∂ ˙ p i ∂p i =∂ ∂p i ∂H ∂q i . Interchanging the derivative these terms cancel one another ( ∇· ~v = 0 in phase space) and we are left with Liouville’s theorem: 0 = ∂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 ﬂuid, but in phase space instead of coordinate space....
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 Fall '11
 berg
 mechanics, Fundamental physics concepts, Vector Motors, phase space

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