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Unformatted text preview: (38) Liouville’s Theorem: We deﬁne the velocity in phase space as a 2 n-dimensional vector ~v = ( ˙ q 1 , . . . , ˙ q n , ˙ p 1 , . . . , ˙ p n ). A large collection of particles can be de-scribed by their density in phase space ρ ( q 1 , . . . , q n , p 1 , . . . , p n , t ). If their are no sources or sinks, we have a conserved current ∂ρ ∂t + ∇ · ( ~v ρ ) = 0 (1) where ∇ = ± ∂ ∂q 1 , . . . , ∂ ∂q n , ∂ ∂p 1 , . . . , ∂ ∂p n ! is the gradient in phase space . (a) Expand (1) in sums of partial derivatives (you get ﬁve terms when you keep coordinates and momenta in separate contributions). (b) Use Hamilton’s equa-tions of motion to show that two terms cancel out. (c) Combine the remaining terms to dρ dt = 0 (Liouville s Theorem) . (2) Due in class (6 points)....
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