Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 3 / 8B (Classical Dynamics)

The kinetic energy of the three unit masses is $\frac{1}{2}{\sum }_i{\dot{x}}_i^2$. Expanding the potential givesThus the small-oscillation mass and stiffness matrices arewhere $J$ is the all-ones matrix.

The vector $(1,1,1{)}^T$ is an eigenvector of the stiffness matrix with eigenvalue $1$. Every vector orthogonal to it has coordinate sum zero and is an eigenvector with eigenvalue $4$. Since $T=I$, the generalized eigenvalue problem for small oscillations gives ${\omega }^2$ equal to these eigenvalues. One orthonormal choice of normal modes isThe last two may be replaced by any orthonormal basis of the sum-zero plane because the frequency is degenerate. Hence the normal frequencies are $\boxed{1,2,2}$.