Singular kinetic matrix in small oscillations ($\mathsf{M}\ddot{q}+\mathsf{K}q=0,\det \mathsf{M}=0$)

If the quadratic kinetic energy has a singular coefficient matrix, some generalized coordinates have no independent linearized inertia. The small oscillation equations can then contain algebraic constraints, and the number of finite normal mode frequencies can be smaller than the number of coordinates. One should use the polynomial $\det (\mathsf{K}-{\omega }^2\mathsf{M})$ without dividing by $\det \mathsf{M}$. Giving a nominally massless component a small mass can restore a high-frequency mode whose frequency diverges as that mass tends to zero.