HW14_prob3_GPS_G37 - Goldstein 9-37 (3rd ed. 9.37) Let r be...

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Goldstein 9-37 (3 rd ed. 9.37) Let r be the position vector of the bob of mass m . In spherical coordinates the components of r are x = r sin θ cos ψ, (1) y = r sin θ sin ψ, (2) z = r cos θ, (3) where r = | r | = constant, θ is the polar angle between r and the z axis, and ψ is the azimuthal angle measured from the x axis in the x - y plane. After transforming into spherical coordinates the Lagrangian reads L = m 2 [ r 2 · θ 2 + r 2 · ψ 2 sin 2 θ ] V ( r, θ ) , (4) where V is the gravitational potential energy of the bob, V = mgz = mgr cos θ. (5) Next, using the prescription, p j = ∂L/∂ · q j ,we f nd the following results for the conjugate momenta p θ and p ψ : p θ = mr 2 · θ, (6) and p ψ = mr 2 · ψ sin 2 θ. (7) Next we construct the energy function h ,de f ned here as h = · θ ∂L · θ + · ψ ∂L · ψ L. (8) We f nd the result h = m 2 [ r 2 · θ 2 + r 2 · ψ 2 sin 2 θ ]+ mgr cos θ. (9) To get the Hamiltonian, we substitute for · θ and · ψ using Eqs.(6) and (7). The result is H = 1 2 m [ p 2 θ r 2 + p 2 ψ r 2 sin 2 θ ]+ mgr cos θ. (10) The point of this exercise so far is to identify the canonical variables θ , ψ , p θ , and p ψ to be used in evaluating the angular momentum Poisson brackets. For example, with these canonical variables the bracket [ L
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HW14_prob3_GPS_G37 - Goldstein 9-37 (3rd ed. 9.37) Let r be...

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