= Equal-endpoint Euclidean oscillator path
{title2=$x_0(\tau)=x\cosh[\omega(\tau-\mathcal T/2)]/\cosh(\omega\mathcal T/2)$}
For positive <mass> $m$ and <angular frequency> $\omega$, the <Euclidean action> $S_E=\tfrac m2\int_0^{\mathcal T}(\dot x^2+\omega^2x^2)\,d\tau$ with $x(0)=x(\mathcal T)=x$ has the unique minimizing path displayed. The <Euler-Lagrange equation> is $\ddot x_0=\omega^2x_0$, and the positive <second variation> proves uniqueness. On this path the integrand is $\tfrac m2\,d(x_0\dot x_0)/d\tau$, giving $S_E[x_0]=m\omega x^2\tanh(\omega\mathcal T/2)$. Increasing the interval of <imaginary time> lets its interior approach the minimum of the <potential energy> even while both endpoints remain fixed.
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