Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 2 / 14D / a / Solution

{title2=$\Phi$}

For three ordered angular positions $q_1,q_2,q_3$, define consecutive gaps
$$
\alpha=q_2-q_1,
\qquad
\beta=q_3-q_2,
\qquad
\gamma=2\pi-q_3+q_1
$$
and the collective angle $\Phi=(q_1+q_2+q_3)/3$. Then $\alpha+\beta+\gamma=2\pi$. For three equal masses on a circle of radius $r$,
$$
T=\frac{3mr^2}{2}\dot\Phi^2
+\frac{mr^2}{3}
\left(\dot\alpha^2+\dot\alpha\dot\beta+\dot\beta^2\right).
$$
Thus the <ignorable coordinate> $\Phi$ decouples from the relative gap dynamics whenever the potential depends only on the gaps.

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 Incoming links: Collective rotation and gap coordinates on a circle