s1005_20

s1005_20 - i + p i p i ! + n X i =1 q i q i + p i p 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 deFned 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
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Unformatted text preview: i + p i p 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 o i p i =- p i H q i . Interchanging the derivative these terms cancel one another and we are left with Liouvilles theorem: = 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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