# Solutions_Part54 - Goldstein 9-37(3rd ed 9.37 Let r be the...

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Goldstein 9-37 (3rded. 9.37)Letrbe the position vector of the bob of massm. In spherical coordinatesthe components ofrarex=rsinθcosψ ,(1)y=rsinθsinψ ,(2)z=rcosθ ,(3)wherer=|r|=constant,θis the polar angle betweenrand thezaxis, andψis the azimuthal angle measured from thexaxis in thex-yplane.Aftertransforming into spherical coordinates the Lagrangian readsL=m2[r2·θ2+r2·ψ2sin2θ]V(r, θ),(4)whereVis the gravitational potential energy of the bob,V=mgz=mgrcosθ .(5)Next, using the prescription,pj=∂L/∂·qj, wefind the following results for theconjugate momentapθandpψ:pθ=mr2·θ ,(6)andpψ=mr2·ψsin2θ .(7)Next we construct the energy functionh, defined here ash=·θ∂L·θ+·ψ∂L·ψL .(8)Wefind the resulth=m2[r2·θ2+r2·ψ2sin2θ] +mgrcosθ .(9)To get the Hamiltonian, we substitute for·θand·ψusing Eqs.(6) and (7).Theresult isH=12m[p2θr2+p2ψr2sin2θ] +mgrcosθ .(10)The point of this exercise so far is to identify the canonical variablesθ,ψ,pθ,andpψto be used in evaluating the angular momentum Poisson brackets.Forexample, with these canonical variables the bracket[Lx, Ly]is computed as[Lx, Ly] =∂Lx∂θ∂Ly∂pθ∂Lx∂pθ∂Ly