s1005_20

# s1005_20 - i ∂ ˙ p i ∂p i n X i =1 ∂ρ ∂q i ˙ q i...

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Solution for assignment 20 set 5 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
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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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