Past exam of the mathematical methods II course of the University of Oxford / 2014 resit / 8 / c

A symmetric quadratic form has twice its off-diagonal entry as the coefficient of the mixed product. Comparing with the preceding potential energy and kinetic energy expressions givesHere $\mathbf{V}$ is the stiffness matrix and $\mathbf{T}$ the mass matrix in angular coordinates; their units differ accordingly. The dimensionless mass-matrix determinant is $8-4=4>0$ and its first diagonal entry is positive, so the kinetic energy is positive for every nonzero angular velocity.