Past exam of the mathematics course of the University of Cambridge / 2004 / ib / Paper 3 / 6D / Solution

For an endpoint-fixed variation $\eta =\delta x$,The boundary term vanishes. Thus every solution of the harmonic oscillator equation is stationary for this action functional. For $\sin \omega T\ne 0$, the endpoint-matching solution is the linear combinationwhich directly satisfies $x_c(0)=a$, $x_c(T)=b$ and ${\ddot{x}}_c=-{\omega }^2{x}_c$. This proves stationarity, rather than assuming the explicit path is a minimum.

On that path, $(x_c{\dot{x}}_c{)}^{\prime }={\dot{x}}_c^2-{\omega }^2{x}_c^2$, so integration by parts reduces the entire action to an endpoint term. The endpoint derivatives areTherefore the fixed-endpoint harmonic-oscillator action isThe displayed formula assumes nonresonant endpoints. If $\omega \ne 0$ and $\omega T=k\pi$, a solution exists only for $b=(-1{)}^{k}a$; then $x=a\cos \omega t+B\sin \omega t$ is a family and its action is zero. For $\omega =0$ the limiting stationary path is linear, with action $(b-a{)}^2/(2T)$.