Coupled oscillator
= 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 $K\mathbf a=\omega^2M\mathbf a$. 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.