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All problems from Goldstein. 1. Show that the function S=(q? + a*)cot(at) - maqa cse (st) is a solution to the HJ equation for the Hamiltonian H = 2m (p+m*w*q*) . Show that the function S generates a correct solution to the harmonic oscillator in time.

All problems from Goldstein. 1. Show that the function S=(q? + a*)cot(at) - maqa cse (st) is a solution to the HJ equation for the Hamiltonian H = 2m (p+m*w*q*) . Show that the function S generates a correct solution to the harmonic oscillator in time.

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All problems from Goldstein. 1. Show that the function S=(q? + a*)cot(ot)- mwqa csc (wt) is a solution to the HJ equation for the Hamiltonian H = 2m (p² + m?w*q?) . Show that the function S generates a correct solution to the harmonic oscillator in time.
Transcribed Image Text:All problems from Goldstein. 1. Show that the function S=(q? + a*)cot(ot)- mwqa csc (wt) is a solution to the HJ equation for the Hamiltonian H = 2m (p² + m?w*q?) . Show that the function S generates a correct solution to the harmonic oscillator in time.
Expert Solution
Step 1

The Hamiltonian-Jacobi equation be defined as,

Hq,∂S∂q+∂S∂t=0

Step 2

Differentiate the given function S with respect to q and t,

S=mω2q2+α2cotωt-mωqαcscωt∂S∂q=mωqcot-mωqαcscωt∂S∂q=-mω2q2+α21sin2ωt+mω2qαcosωtsin2ωt

Step 3

Now substitute the above values in the left side of equation 1

Hq,∂S∂q+∂S∂t=12m∂S∂q2+mω2q2+∂S∂t=12mm2ω2sin2ωtqcosωt-α2+12mω2q2-12mω2q2+ω2sin2ωt+mω2qαcosωtsin2ωt=12mω2q2cos2ωt+α2sin2ωt-mω2qαcosωtsin2ωt+12mω2q2-12mω2q2+ω2sin2ωt+mω2qαcosωtsin2ωt=0

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