= Solution
Lift the circle to its real-valued angular covering coordinate. The free classical equation is $\ddot\theta=0$. For each <winding number> $p\in\mathbb Z$, the endpoint lift is $\theta+2\pi p$, and
$$
\boxed{\theta_p(s)=\theta'+\frac{\Delta+2\pi p}{T}(s-t'),\qquad
S_p=\frac{I(\Delta+2\pi p)^2}{2T}}.
$$
Positive and negative windings are distinct <classical paths> with the same physical endpoints. Reducing the angle modulo $2\pi$ returns their trajectories on the ring.
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