 Coupled oscillator

Coupled oscillators influence one another's motion through shared forces or other interactions. Two pendulums connected by a light spring provide a visible example: moving either bob changes the force on the other. For equal masses and lengths, the symmetric motion leaves the connecting spring unstretched, while the antisymmetric motion stretches it and has a higher frequency.

For a conservative linear system, the equations of motion take the form $M\ddot{\mathbf{q}}+K\mathbf{q}=0$, where $M$ is the mass matrix and $K$ the elastic stiffness matrix. A normal mode has a fixed displacement pattern $\mathbf{a}$ satisfying $K\mathbf{a}={\omega }^{2}M\mathbf{a}$. Expanding the initial motion in normal modes separates the evolution. Displacing just one pendulum generally excites several normal modes, producing beats and visible transfer of energy. A driving force excites only the modes onto which its spatial pattern has a nonzero projection.