= Fixed-endpoint harmonic-oscillator action
{title2=$S_{\rm cl}=\frac{\omega[(a^2+b^2)\cos\omega T-2ab]}{2\sin\omega T}$}
For $S=\tfrac12\int_0^T(\dot x^2-\omega^2x^2)dt$ and fixed endpoint values $a,b$, the <Euler-Lagrange equation> is $\ddot x+\omega^2x=0$. When $\sin\omega T\ne0$, solve the endpoint equations for the unique trajectory and use $\dot x^2-\omega^2x^2=(x\dot x)'$ on that trajectory to obtain the displayed <action>. For $\omega\ne0$, at $\omega T=k\pi$ a solution exists only if $b=(-1)^ka$; then there is a family of solutions and their action is zero. At $\omega=0$ the nonresonant formula has the limit $(b-a)^2/(2T)$.
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