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Unformatted text preview: (38) Liouville’s Theorem: We deﬁne the velocity in phase space as a 2 ndimensional vector ~v = ( ˙ q 1 , . . . , ˙ q n , ˙ p 1 , . . . , ˙ p n ). A large collection of particles can be described 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 equations 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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This note was uploaded on 12/11/2011 for the course PHY 4936 taught by Professor Berg during the Fall '11 term at FSU.
 Fall '11
 berg
 mechanics, Momentum, Work

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