 Solution

ID: past-exam-of-the-theoretical-physics-i-course-of-the-university-of-cambridge/2022/1/d/solution

I_{1} \ddot{θ} = - κ sin (θ - φ), I_{2} \overset{φ}{¨} = κ sin (θ - φ) .

sin (θ - φ) = 0

. Hence the complete equilibrium families, with angles taken modulo

2 π

, are

(θ, φ) = (q, q) or (θ, φ) = (q + π, q), q \in R / (2 π Z) .

The first family is aligned in azimuth and minimizes the spring length; the second is antipodal and maximizes it. Their potentials are

k [(a - A)^{2} + h^{2}] /2

and

k [(a + A)^{2} + h^{2}] /2

, respectively. Common rotations generate a whole family rather than isolated equilibria.

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