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A11f09 - (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

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PHY 4938 HOME AND CLASS WORK – SET 9 (November 26, 2011) Read Landau-Lifshitz § 40 and § 42. (36-A) Legendre transformation: Deﬁne the Hamiltonian by H = X j ˙ q j ∂L ˙ q j - L and the generalized momentum by p j = ∂L ˙ q j . Show that the Hamiltonian is a function of q j and p j only: H = H ( q j , p j ). Then derive Hamilton’s equations of motion. Hint: Calculate dH . Due in class (4 points). (36-B) Problem 1, Landau-Lifshitz § 40, p.133. Show ﬁrst: H = T + U . Due in class (4 points). (36-C) Derive Newton’s equation from Hamilton’s equation for the Cartesian case. Due in class (2 points). (37) Poisson brackets are deﬁned by [ g, h ] = X k ± ∂g ∂q k ∂h ∂p k - ∂h ∂q k ∂g ∂p k ! where g and h are functions of q i , p i and, possibly, t . Show the following properties (due December 7 before class, 10 points): 1. dg dt = [ g, H ] + ∂g ∂t . 2. ˙ q j = [ q j , H ] . 3. ˙ p j = [ p j , H ] . 4. [ x i , x j ] = [ p i , p j ] = 0 ; [ x i , p j ] = δ ij , 5. [ x i , L j ] = ± ijk x k , [ p i , L j ] = ± ijk p k , and [ L i , L j ] = ± ijk L k , , where the Einstein summation convention is used and L j = ± jkl x k p l is the j th component of the angular momentum of the system.

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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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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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A11f09 - (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

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