Simulate simple and double pendulums with adjustable length, mass, and damping. Shows phase space plots and chaotic motion.
Pendulum Simulator numerically integrates the nonlinear equation of motion for a simple pendulum (bob on a rigid, massless rod) or the coupled equations for a double pendulum (two linked rods and masses), rather than relying on the textbook small-angle formula. You set length, bob mass, starting angle, and a damping coefficient, then watch an angle-vs-time trace and a phase-space plot (angular velocity against angle) update as the simulation runs. The common surprise: push the starting angle past roughly 20-30 degrees and the measured period grows noticeably longer than 2 pi times the square root of L over g, because that formula only holds in the small-angle limit.
A concrete input and expected output from the current implementation.
Input
Simple pendulum, length = 1 m, gravity = 9.81 m/s^2, starting angle = 5 degrees, damping = 0
Expected output
Period ~= 2.006 s
At a 5-degree starting angle the motion stays in the near-linear regime, so the numerically integrated period matches the small-angle formula 2*pi*sqrt(L/g) = 2*pi*sqrt(1/9.81) to within about a millisecond.
For the simple pendulum the tool solves theta'' = -(g/L)*sin(theta) - b*theta' step by step, where b is the damping coefficient, instead of substituting sin(theta) with theta. For the double pendulum it solves the four coupled first-order equations from the two-mass Lagrangian, advancing theta1, theta2, and both angular velocities together so the arms stay physically coupled at every step.
Alongside the angle-vs-time trace, the simulator plots angular velocity against angle. An undamped simple pendulum traces a closed loop repeating every period; adding damping spirals that loop inward to the origin as energy dissipates. In double-pendulum mode the equivalent plot fills in an irregular, non-repeating path - the visual signature of chaotic motion, not a rendering glitch.
No, not for the simple pendulum - in this model mass cancels out of the equation of motion, so the period depends only on length, gravity, and starting angle. In double-pendulum mode the mass of each arm does change the motion, because the two arms are dynamically coupled through their masses.
The double pendulum is chaotic: any difference in starting angle too small to notice on the input, or in the integrator's step size, grows exponentially over time. If both runs used exactly the same entered values they should match; if the paths diverge after a few swings, that is expected chaotic behavior, not an error.
The example and behavioral notes were checked against the browser implementation. Standards and primary references below define the relevant format, formula, or platform behavior.
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