The 3D Spherical Pendulum: Rosettes, Cones and Foucault Precession

A bob free to swing in two directions

A simple pendulum swings in a single plane. A spherical pendulum removes that constraint: the bob hangs from a rigid rod or an inextensible string of fixed length and is free to move anywhere on the surface of a sphere centred on the pivot. It has two independent angular degrees of freedom — a polar angle θ measuring how far it has swung from straight down, and an azimuthal angle φ measuring its rotation around the vertical axis — instead of the one angle of a planar pendulum.

Two conserved quantities shape every orbit

Two things are conserved as the bob swings, and together they explain everything the trajectory can do. Total mechanical energy is conserved because gravity is the only force doing work and it is conservative. Angular momentum about the vertical axis is also conserved, because gravity's torque about that axis is zero — the pendulum's weight always points straight down, through the axis, so it cannot twist the bob around it. That second conservation law is the key difference from the planar case: any sideways push imparts an angular momentum about the vertical axis, and unless the bob is launched with exactly zero of it, it can never again pass through the vertical plane it started in.

Rosettes, and the special case of a perfect cone

With generic initial conditions the bob traces an open, precessing path that fills an annular band on the sphere, swinging between a minimum and maximum polar angle while its azimuthal angle keeps advancing — a rosette. Viewed from directly above, the bob's horizontal projection looks like a flower with the same number of petals appearing over and over as the two periods, the swing period and the precession period, drift in and out of a simple ratio. There is one special launch condition where θ never changes at all: launch the bob purely horizontally with exactly the speed that makes gravity's downward pull equal the centripetal force needed for uniform circular motion at that angle. Then the bob traces a perfect horizontal circle at constant height, and the whole system is a conical pendulum — the rod sweeps out a cone, and the period depends only on the cone's half-angle and the pendulum length, not on the mass.

conical pendulum period:  T = 2π √( L cosθ / g )
   L = rod length, θ = angle from vertical, g = gravitational acceleration
   (independent of mass — the same cancellation as in a simple pendulum's period)

Integrating it: why RK4

The equations of motion for θ and φ are coupled, nonlinear second-order ODEs — there is no closed-form solution for the general rosette, only for the small handful of special cases like the conical pendulum. This simulation integrates them numerically with fourth-order Runge–Kutta, sampling the derivative four times per step and blending the samples, which gives high per-step accuracy for a system that is not being run for millions of steps, so RK4's lack of exact energy conservation over very long runs is not the limiting concern it would be for, say, a planetary orbit simulation.

Foucault mode: when the pivot itself is rotating

Switch the simulation to Foucault mode and the pivot is no longer inertial — it is a point on the rotating Earth. In the pivot's rotating reference frame an extra fictitious force appears, the Coriolis force, and it very slowly precesses the pendulum's swing plane even when the pendulum is launched as a purely planar oscillation. The precession rate is Ω·sin(latitude), where Ω is Earth's rotation rate: at the poles the swing plane rotates once per sidereal day (sin 90° = 1), at the equator it does not precess at all (sin 0° = 0), and at mid-latitudes it precesses at the intermediate rate that Léon Foucault first used in 1851 to give the public a direct, mechanical demonstration that the Earth itself rotates, without looking at the sky.

Frequently asked questions

Try it live

Everything above runs in your browser — open 3D Spherical Pendulum and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.