Answered: Show that the function S = (mω/2)(q2 +…

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Show that the function

S = (mω/2)(q² + α²) cot ωt − mωqα csc ωt
is a solution of the Hamilton-Jacobi for Hamilton’s principal function for the
linear harmonic oscillator with

H =(1/2m)(p² + m²ω²q²).

Show that this function generates a correct solution to the motion of the
harmonic oscillator.

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Show that the function \[ S=S(q, \beta, t)=\frac{m \omega}{2}\left(q^{2}+\beta^{2}\right) \cot \omega t-m \omega q \beta \csc \omega t \] is a solution to the Hamilton-Jacobi equation for Hamilton's principal function for the linear harmonic oscillator with \[ H=\frac{1}{2 m}\left(p^{2}+m^{2} \omega^{2} q^{2}\right) \] Show that this function generates a correct solution to the motion of the harmonic oscillator.
Provide a written answer