Fixed-endpoint harmonic-oscillator action ($S_{\mathrm{cl}}=\frac{\omega [(a^2+b^2)\cos \omega T-2ab]}{2\sin \omega T}$)

For $S=\frac{1}{2}{\int }_0^T({\dot{x}}^2-{\omega }^2{x}^2)dt$ and fixed endpoint values $a,b$, the Euler-Lagrange equation is $\ddot{x}+{\omega }^{2}x=0$. When $\sin \omega T\ne 0$, solve the endpoint equations for the unique trajectory and use ${\dot{x}}^2-{\omega }^2{x}^2=(x\dot{x}{)}^{\prime }$ on that trajectory to obtain the displayed action. For $\omega \ne 0$, at $\omega T=k\pi$ a solution exists only if $b=(-1{)}^{k}a$; then there is a family of solutions and their action is zero. At $\omega =0$ the nonresonant formula has the limit $(b-a{)}^2/(2T)$.