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Collective rotation and gap coordinates on a circle ( $Φ$ )

For three ordered angular positions

q_{1}, q_{2}, q_{3}

, define consecutive gaps

α = q_{2} - q_{1}, β = q_{3} - q_{2}, γ = 2 π - q_{3} + q_{1}

and the collective angle

Φ = (q_{1} + q_{2} + q_{3}) /3

. Then

α + β + γ = 2 π

. For three equal masses on a circle of radius

r

,

T = \frac{3 m r ^{2}}{2} \dot{Φ}^{2} + \frac{m r ^{2}}{3} (\overset{α}{˙}^{2} + \overset{α}{˙} \dot{β} + \dot{β}^{2}) .

Thus the ignorable coordinate

Φ

decouples from the relative gap dynamics whenever the potential depends only on the gaps.

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