Spherical Pendulum — Top-Down Rosette (2D)

Bird's-eye view of the bob's precessing path, integrated from the real (θ,φ) equations of motion.

Launch

Initial polar angle θ₀ 30° Initial azimuthal velocity φ̇₀ 1.20 Pendulum length L 1.00 Damping k 0.00

Show trace (rosette)

Live state

θ (from vertical)—

φ (azimuth)—

θ̇—

φ̇—

L_z (conserved)—

Energy E—

About

This is not a simple back-and-forth planar pendulum: the bob is free to swing in any direction, so viewed from above its path is a precessing ellipse/rosette rather than a straight line. The equations come directly from the Lagrangian L = ½mL²(θ̇² + sin²θ·φ̇²) + mgL·cosθ, giving θ̈ = sinθcosθ·φ̇² − (g/L)sinθ and a conserved vertical angular momentum L_z = mL²sin²θ·φ̇ (undamped). Try φ̇₀=0 to see the rosette degenerate into a straight-line planar swing, and increase it to see genuine precession.