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HW14_prob3_GPS_G37

# 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 fi 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 fi ned here as h = · θ ∂L · θ + · ψ ∂L · ψ L . (8) We fi 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 x , L y ] is computed as [ L x , L y ] = ∂L x ∂θ ∂L y ∂p θ ∂L x ∂p θ ∂L y

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

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