a1005 - dρ dt = 0(Liouville s Theorem(2 Due February 5 in...

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ADVANCED DYNAMICS — PHY 4241/5227 HOME AND CLASS WORK – SET 5 (February 1, 2010) (20) Liouville’s Theorem: We defne the velocity in phase space as a 2 n -dimensional vector ~v = ( ˙ q 1 , . . . , ˙ q n , ˙ p 1 , . . . , ˙ p n ). A large collection o± particles can be de- scribed by their density in phase space ρ ( q 1 , . . . , q n , p 1 , . . . , p n ). I± 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 o± partial derivatives (you get fve terms when you keep coordinates and momenta in separate contributions). (b) Use Hamilton’s equa- tions o± motion to show that two terms cancel out. (c) Combine the remaining terms to
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Unformatted text preview: dρ dt = 0 (Liouville s Theorem) . (2) Due February 5 in class (6 points). Read M&T chapter 7.12. (21) Poisson brackets are defned by [ g, h ] = X k ∂g ∂q k ∂h ∂p k-∂h ∂q k ∂g ∂p k ! where g and h are ±unctions o± q i , p i and, possibly, t . Show the ±ollowing properties (due February 10 be±ore 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 ] = ² ijk x k , [ p i , L j ] = ² ijk p k , and [ L i , L j ] = ² ijk L k , , where the Einstein summation convention is used and L j = ² jkl x k p l is the i th component o± the angular momentum o± the system....
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