 Normal coordinate

A normal coordinate measures motion along a particular normal mode of a linearized mechanical system. For small displacements $\mathbf{q}$ about a stable equilibrium, write the kinetic energy and potential energy as quadratic forms with mass matrix $M$ and stiffness matrix $K$. Mode vectors satisfy the generalized eigenvalue problemFor a conservative system with positive definite $M$ and symmetric $K$, mass-orthonormal mode vectors transform the coupled equations of motion into independent equations ${\ddot{Q}}_j+{\omega }_j^2{Q}_j=0$.

Two equal coupled oscillators give a simple example: their sum coordinate describes motion together, and their difference coordinate describes opposite motion. Exciting only one of these patterns leaves the other coordinate zero in the linear approximation. This is useful for interpreting measured resonances and molecular vibrations; large displacements or nonlinear interactions can couple the modes again.