 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 areThese exhaust the constant-velocity paths in the different winding sectors.