HW14_prob4_Pbfortop

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

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Poisson Brackets for the Heavy Top The f rst thing to do is de f 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 f nition, p j = ∂L/∂ · q j . Two of these, p ψ and p φ ,areg ivenbyEqs . (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 f 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 .
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Unformatted text preview: taining only the nonzero terms. Note that since the function f ( , , ) does not depend on the canonical momenta, we can immediately write [ , f ( , , )] = X j p j f q j , (2) and since does not depend on p , this simpli f es to [ , f ( , , )] = p f + p f . (3) The partial derivatives of are easy to calculate p = cot I 1 sin , (4) and p = 1 I 3 + cot 2 I 1 . (5) Putting the pieces together, we obtain [ , f ( , , )] = cot I 1 sin f 1 I 3 + cot 2 I 1 f . (6) 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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