 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/i/solution

The four spring extensions are

x_{1}

x_{2} - x_{1}

x_{3} - x_{2}

and

- x_{3}

k [x_{1}^{2} + (x_{2} - x_{1})^{2} + (x_{3} - x_{2})^{2} + x_{3}^{2}] /2

, while the kinetic energy is

m \sum_{i} \overset{x}{˙}_{i}^{2} /2

. Thus

L = \frac{m}{2} i = 1 \sum 3 \overset{x}{˙}_{i}^{2} - \frac{k}{2} [x_{1}^{2} + (x_{2} - x_{1})^{2} + (x_{3} - x_{2})^{2} + x_{3}^{2}],

with the matrices in the requested quadratic form

T = m I_{3}, V = k 2 - 1 0 - 1 2 - 1 0 - 1 2 .

The equilibrium offsets disappear because the expansion is in displacements from static equilibrium.

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