 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2021/paper-2/7/a/iii/solution

Using the three real eigenvectors and positive frequencies in part (ii), the general motion is

x (t) = j = 1 \sum 3 u_{j} [A_{j} cos (ω_{j} t) + B_{j} sin (ω_{j} t)] .

There are six independent real constants,

A_{1}, A_{2}, A_{3}, B_{1}, B_{2}, B_{3}

, corresponding to the three initial displacements and three initial velocities. With the displayed orthonormal eigenvectors,

A_{j} = u_{j}^{T} x (0)

and

B_{j} = u_{j}^{T} \dot{x} (0) / ω_{j}

. This spans all solutions because the normal-mode eigenvectors form a basis.

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