3D Ornament Pendulum Chaos

What is modelled

Two glass ornaments hang from a fir branch on thin strings: the upper one from the branch, the lower one from the upper ornament. The strings are massless and inextensible, the ornaments are point masses, and the motion stays in one vertical plane. This is the classic double pendulum, which is regular for small swings and chaotic for large ones. A ghost twin starts a tiny angle away so you can watch nearby trajectories separate.

Equations

• Lagrangian with angles θ₁, θ₂ from the downward vertical: T = ½(m1+m2)L1²ω1² + ½m2L2²ω2² + m2L1L2ω1ω2cos(θ1−θ2), U = −(m1+m2)gL1cosθ1 − m2gL2cosθ2.
• Equations of motion as a 2×2 linear system M α = f with M = [[(m1+m2)L1², m2L1L2cosΔ],[m2L1L2cosΔ, m2L2²]], solved at every Runge-Kutta stage (RK4, step 1/480 s). Joint friction adds the torques −cω1 + c(ω2−ω1) and −c(ω2−ω1).
• Small oscillations: normal-mode frequencies are the roots of det(K − ω²M) = 0; for equal masses and lengths ω² = (2 ± √2) g / L.
• Chaos: nearby trajectories separate as d(t) ≈ d0 eλt. The largest Lyapunov exponent λ is estimated with the Benettin method: a second copy of the system is pulled back to distance d₀ every 0.05 s and the average of ln(d/d₀) per unit time is reported. The predictability time is about ln(1/d0)/λ.

Checks run in the page

• The mass-matrix accelerations are compared with the textbook closed-form expressions at 400 random states and parameter sets.
• With friction off, the total energy T + U of the current chaotic motion must stay constant over 20 s.
• Small-amplitude runs started in each normal mode must oscillate with the periods from the linearised eigenproblem.

Honest simplifications

• Real baubles are extended bodies that spin, and the string may go slack or touch the branch; none of that is modelled. The ornaments are drawn larger than the point masses, and the lower one scales with its mass.
• Air drag is replaced by joint friction (a viscous torque). Real drag is quadratic in speed.
• The Lyapunov value is a finite-time estimate and depends on the trajectory; for regular motion it tends to zero.
• Chaos here is deterministic: the same start always gives the same path, while the numerical error of any integrator eventually decouples a long run from the true solution.

What to try

• Compare the small-swing preset with the chaotic drop and watch the ghosts separate or not.
• Start in a normal mode (slow, in phase; fast, opposite) and see the regular motion.
• Turn on friction and watch the energy fall until the baubles rest.