Past exam of the mathematical methods II course of the University of Oxford / 2021 resit / 11 / c / Solution

Define a common displacement coordinate $X=(x_1+q{x}_2)/(1+q)$ and a relative displacement $\delta =x_1-x_2$. Adding and subtracting the equations of motion givesThe general real motions of the two normal modes are thereforeHere $C,D,A,B$ are independent real constants. The zero mode includes uniform drift as well as a static common displacement; writing only a constant times $\cos 0t$ would miss the independent linear-in-time solution. Their superposition isIn the oscillatory normal mode, $m_1{x}_1+m_2{x}_2=0$ and the two amplitudes have ratio $1/q$, with opposite phases. Equivalently the oscillatory motion has zero total angular momentum about the hoop center.

As $q\to \infty$ with $m_1,k$ fixed, $\omega \to \sqrt{2k/m_1}$ and the heavier particle's oscillatory amplitude tends to zero for fixed lighter-particle amplitude. The lighter particle then oscillates against two effectively anchored springs with combined spring constant $2k$. A common uniform rotation remains an allowed zero mode; the heavy particle is stationary in the limit only if the initial common drift is zero or tends to zero, as it does for bounded total angular momentum.