{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

HW14_prob1_GPS30_G28

# HW14_prob1_GPS30_G28 - Goldstein 9-28(3rd ed 9.30(a Let H...

This preview shows pages 1–3. Sign up to view the full content.

Goldstein 9-28 (3 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 ff erentiation. If we compare Eq.(8) with Eqs.(3) and (4), we see that the desired result is established. 1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
(b) If the Hamiltonian H and the function F are constants of the motion, then it must be true that dF dt = F t + [ F, H ] = 0 , (9) and F t = [ H, F ] . (10) Since, H and F are each constants of the motion, we know from part (a) that [ H, F ] is also a constant of the motion, and from Eq.(10) it is, thus, clear that
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 3

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online