Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2021 / Paper 2 / 7 / b / i

Table of contents
- Solution i

Solution

Introduce the fixed endpoint values

x_{0} = x_{N + 1} = 0

. The Lagrangian is

L = \frac{m}{2} j = 1 \sum N \overset{x}{˙}_{j}^{2} - \frac{k}{2} j = 0 \sum N (x_{j + 1} - x_{j})^{2} .

Thus

T = m I_{N}

,

V_{ij} = k (2 δ_{ij} - δ_{i, j + 1} - δ_{i, j - 1})

, and

V - ω^{2} T = d - k 0 ⋮ 0 - k d - k ⋱ \dots 0 - k d ⋱ 0 \dots ⋱ ⋱ ⋱ - k 0 ⋮ 0 - k d, d = 2 k - m ω^{2} .

This is a symmetric tridiagonal matrix, reflecting that only nearest-neighbour masses are coupled.

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