HW14_prob1_GPS30_G28

HW14_prob1_GPS30_G28 - Goldstein 9-28 (3rd ed. 9.30) (a)...

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rd ed. 9.30) (a) Let H be a Hamiltonian for some dynamical system. If A and B are two constants of the motion that explicitly depend on time, then it must be true that A t = [ A,H ]=[ H,A ] , (1) and B t = [ B,H ]=[ H,B ] . (2) We want to prove that [ A,B ] is also a constant of the motion, or that C t +[ C,H ]=0 , (3) where C =[ A,B ] . (4) For the three functions A , B ,and H , we can use Jacobi’s identity to write [[ A,B ] ,H ]+[[ B,H ] ,A ]+[[ H,A ] ,B ]=0 , (5) which we can rewrite using Eqs.(1) and (2) as [[ A,B ] ,H ]+[ B t ,A ]+[ A t ,B ]=0 . (6) This can immediately be rewritten as [[ A,B ] ,H ]+[ A, B t ]+[ A t ,B ]=0 , (7) and again as [[ A,B ] ,H ]+ [ A,B ] t =0 , (8) by applying the rules of partial di f erentiation. If we compare Eq.(8) with Eqs.(3) and (4), we see that the desired result is established. 1
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This note was uploaded on 12/19/2010 for the course PHYSICS ph 409 taught by Professor Wilemski during the Fall '10 term at Missouri S&T.

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HW14_prob1_GPS30_G28 - Goldstein 9-28 (3rd ed. 9.30) (a)...

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