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s1005_21

# s1005_21 - Solution for assignment 21 set 5 Poisson...

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Solution for assignment 21 set 5 Poisson Brackets (21a) Let us consider functions g = g ( q k , p k , t ) and h = h ( q k , p k , t ). The Poisson bracket is defined by [ g, h ] def = X k ∂g ∂q k ∂h ∂p k - ∂h ∂q k ∂g ∂p k ! . The properties of the assignment are shown in the following: 1. Resulting in the definition of the total time derivative: dg dt = [ g, H ] + ∂g ∂t = X k ∂g ∂q k ∂H ∂p k - ∂H ∂q k ∂g ∂p k ! + ∂g ∂t = X k ∂g ∂q k ˙ q k + ∂g ∂p k ˙ p k ! + ∂g ∂t . 2. Using ∂q j /∂p k = 0: ˙ q j = [ q j , H ] = X k ∂q j ∂q k ∂H ∂p k - ∂H ∂q k ∂q j ∂p k ! = X k δ jk ˙ q k = ˙ q j , 3. Using ∂p j /∂q k = 0: ˙ p j = [ p j , H ] = X k ∂p j ∂q k ∂H ∂p k - ∂H ∂q k ∂p j ∂p k ! = X k δ jk ˙ p k = ˙ p j , 4. Commutators of generalized coordinates and momenta: [ q i , q j ] = X k ∂q i ∂q k ∂q j ∂p k - ∂q j ∂q k ∂q i ∂p k ! = 0 as ∂q j /∂p k = 0 and ∂q i /∂p k = 0. Similarly [ p i , p j ] = X k ∂p i ∂q k ∂p j ∂p k - ∂p j ∂q k ∂p i ∂p k !

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s1005_21 - Solution for assignment 21 set 5 Poisson...

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