s11f1038

s11f1038 - i ! + n X i =1 q i q i + p i p i ! . Using...

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Solution for assignment 38 Liouville’s Theorem We consider motion of point particles with n degrees of freedom in phase space , which is described by a Hamiltonian H ( q 1 , . . . , q n ; p 1 , . . . , p n ) . Let ρ ( q 1 , . . . , q n ; p 1 , . . . , p n ; t ) be the density in phase space and the velocity of the density element is the vector ~v = ( ˙ q 1 , . . . , ˙ q n ; ˙ p 1 , . . . , ˙ p n ) . The gradient is now also defined in phase space (ˆ q i and ˆ p i are unit vectors): = n X i =1 ± ˆ q i ∂q i + ˆ p i ∂p i ! . The continuity equation reads 0 = ∂ρ ∂t + ∇ · ~ j = ∂ρ ∂t + ∇ · ( ~vρ ) as ~ j = ~vρ holds. Therefore, 0 = ∂ρ ∂t + ρ ∇ · ~v + ~v · ∇ ρ = ∂ρ ∂t + ρ n X i =1 ± ˙ q i ∂q i + ˙ p i ∂p
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Unformatted text preview: i ! + n X i =1 q i q i + p i p i ! . Using Hamiltons 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 Liouvilles theorem: 0 = t + n X i =1 q i q i + p i p i ! = d 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 4936 taught by Professor Berg during the Fall '11 term at FSU.

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