Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2019 / Paper 2 / 7 / c

Use the mode shapes from part (b), and put $\omega =\sqrt{2k/m}$. The initial velocity decomposes asThere is no antisymmetric component because the initial state treats the equal masses identically. Initial displacements are zero, so the zero mode contributes a linear drift and the nonzero normal mode contributes a sine:In terms of actual angular positions,The conserved angular momentum is initially $2m\Omega$ and the total moment of inertia is $4m$, so the common angular velocity must be $\Omega /2$, agreeing with the drift. The relative displacements remain of order $\Omega /\omega$, even when the common angle becomes large. Hence small $\Omega /\omega$ preserves the cyclic ordering and small spring extensions at arbitrarily late times. In the ideal arc-spring model the displayed energies are exactly quadratic in the relative angles, so this motion remains exact while that ordering persists.