Small oscillations about equilibrium

Summary

The theory of small oscillations describes the motion of a mechanical system near a stable equilibrium by linearizing its equations of motion. One keeps the kinetic and the potential energy to second order in the velocities and the displacements. This gives a linear system $M \ddot x + K x = 0$ with constant symmetric matrices. When $K$ is positive definite, every solution is a superposition of normal modes, each a simple harmonic motion. This is the starting point of the theory of vibrations of molecules, structures and coupled oscillators. See the definition.