An inverted pendulum has its center of mass above its pivot. For a point mass on a massless rod of length , let measure departure from the upward vertical. The gravitational potential energy is , and the equation of motion is .
For small , linearization gives exponentially growing and decaying solutions with rate : the upright equilibrium is unstable. A moving support or an applied torque can stabilize it through feedback control. Balancing a rod on a hand makes this visible: the hand continually moves the pivot to counter the growing departure.
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An inverted pendulum is a classic problem in physics and engineering that involves a pendulum which is attached to a pivot point above its center of mass. This system is unstable because, unlike a regular pendulum that swings down into a stable equilibrium position, an inverted pendulum is in a natural unstable equilibrium when it is perfectly vertical. The dynamics of an inverted pendulum can be described using principles of mechanics.