Equal-endpoint Euclidean oscillator path ($x_0(\tau )=x\cosh [\omega (\tau -\mathcal{T}/2)]/\cosh (\omega \mathcal{T}/2)$)

For positive mass $m$ and angular frequency $\omega$, the Euclidean action $S_E=\frac{m}{2}{\int }_0^{\mathcal{T}}({\dot{x}}^2+{\omega }^2{x}^2)d\tau$ with $x(0)=x(\mathcal{T})=x$ has the unique minimizing path displayed. The Euler-Lagrange equation is ${\ddot{x}}_0={\omega }^2{x}_0$, and the positive second variation proves uniqueness. On this path the integrand is $\frac{m}{2}d(x_0{\dot{x}}_0)/d\tau$, 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.