physics//mechanics//pendulum
A pendulum is a mass that swings on a rod or string about a pivot under gravity, and it is the canonical example of a nonlinear system with two equilibria of opposite character: hanging, it oscillates and is the model of a crane's swinging load or a drone's slung cargo; upside down, it falls over unless something balances it, and it is the model of a balancing robot, a rocket on its gimbal, or a person standing. Without friction its angle obeys
A pendulum is a mass that swings on a rod or string about a pivot under gravity, and it is the canonical example of a nonlinear system with two equilibria of opposite character: hanging, it oscillates and is the model of a crane's swinging load or a drone's slung cargo; upside down, it falls over unless something balances it, and it is the model of a balancing robot, a rocket on its gimbal, or a person standing. Without friction its angle obeys
θ¨=−gLsinθ.\ddot\theta=-\frac{g}{L}\sin\theta .θ¨=−Lgsinθ.
The same equation gives both behaviours, and the linearization at each equilibrium tells them apart. Near θ∗=0\theta^*=0θ∗=0 the sine is close to the angle and the pendulum oscillates at g/L\sqrt{g/L}g/L: 3.13 rad/s for a one-metre rod, a period of 2 s, independent of the mass. Near θ∗=π\theta^*=\piθ∗=π the cosine in the Jacobian flips sign, the equilibrium becomes a saddle, and a small error grows as e3.13te^{3.13t}e3.13t, doubling every 0.22 s. Same equation, two equilibria, opposite fates; linearization is local on purpose.
A broom balances on a palm more easily than a pencil does.
The rate at which an inverted pendulum falls is g/L\sqrt{g/L}g/L<path d="M263,681c0.7,0,18,39.7,52,119
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M1001 80h400000v40h-400000z"/>, so a long pole topples slowly enough for a hand to keep up, while a short one falls faster than reflexes can follow; the same arithmetic tells a robot designer how fast the balance loop must run.
Balancing it is the classic control benchmark. A cart that moves under the pendulum (the cart-pole) is the textbook unstable plant on which LQR, pole placement and reinforcement learning are first tried, because the loop must be faster than the doubling time and the linearized model holds only near the top (small-angle approximation).
The swing has its own uses and costs. A crane's hook load and a load hanging under a drone form a hanging pendulum whose natural frequency depends on the cable length alone, so anti-sway controllers shape their motion to avoid exciting it; shortening the cable raises the frequency and speeds up the swing.
Away from small angles the period grows and the oscillation is no longer sinusoidal; with enough energy the motion turns into full rotations. The phase portrait shows the closed orbits near the bottom, the saddle at the top and the curve separating swinging from spinning.
Two pendulums linked end to end make the double pendulum, the classic chaotic machine: two releases a millionth of a radian apart end up in completely different places within seconds (chaos).