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.

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.