= Solution
The <Euler-Lagrange equation> gives $\ddot\theta=0$. Lift the path to the real line beginning at $\theta_i$. Its endpoint can be any $\theta_f+2\pi m$, where $m\in\mathbb Z$ labels the relative <winding number>. With $\Delta=\theta_f-\theta_i$, every classical path and its <action> are
$$
\boxed{\theta_m(t)=\theta_i+\frac{\Delta+2\pi m}{T}t,\qquad S_m=\frac{\Lambda}{2T}(\Delta+2\pi m)^2.}
$$
These exhaust the constant-velocity paths in the different winding sectors.
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