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HW14_prob4_Pbfortop

# HW14_prob4_Pbfortop - taining only the nonzero terms Note...

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Poisson Brackets for the Heavy Top The fi rst thing to do is de fi ne the canonical variables, i.e., the generalized coordinates and momenta for the problem. The Euler angles θ , φ , and ψ are the generalized coordinates. The corresponding canonical momenta are found using the de fi nition, p j = ∂L/∂ · q j . Two of these, p ψ and p φ , are given by Eqs. (5-53) and (5-54) of Goldstein. The third one, p θ , is easily found by direct calculation, but we don’t need an explicit form for it to fi nish the problem. To evaluate the given Poisson bracket, we must express · ψ in terms of the canonical variables. This is readily accomplished using Eqs. (5-53), (5-54), and (5-58) of Goldstein. The result is · ψ = p ψ I 3 cos θ p φ p ψ cos θ I 1 sin 2 θ . (1) Now it’s just a matter of substituting Eq.(1) into the PB prescription and re-
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Unformatted text preview: taining only the nonzero terms. Note that since the function f ( θ , φ, ψ ) does not depend on the canonical momenta, we can immediately write [ · ψ, f ( θ, φ, ψ )] = − X j ∂ · ψ ∂p j ∂f ∂q j , (2) and since · ψ does not depend on p θ , this simpli f es to [ · ψ, f ( θ, φ, ψ )] = − ∂ · ψ ∂p φ ∂f ∂φ + ∂ · ψ ∂p ψ ∂f ∂ψ . (3) The partial derivatives of · ψ are easy to calculate ∂ · ψ ∂p φ = − cot θ I 1 sin θ , (4) and ∂ · ψ ∂p ψ = 1 I 3 + cot 2 θ I 1 . (5) Putting the pieces together, we obtain [ · ψ, f ( θ, φ, ψ )] = μ cot θ I 1 sin θ ∂f ∂φ − ½ 1 I 3 + cot 2 θ I 1 ¾ ∂f ∂ψ ¶ . (6) 1...
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