Mathematical Model: The Double Pendulum The double pendulum is a classic example of a nonlinear dynamical system exhibiting deterministic chaos. To model its motion, we use Lagrangian Mechanics, focusing on energy states rather than force vectors. 1. Generalized Coordinates The system is defined by two angles, θ₁ and θ₂, measured from the vertical. The positions of the masses (m₁, m₂) are: • Mass 1: x₁ = L₁ sin(θ₁), y₁ = -L₁ cos(θ₁) • Mass 2: x₂ = x₁ L₂ sin(θ₂), y₂ = y₁ - L₂ cos(θ₂) 2. Kinetic (

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Chaotic Double Ring Pendulum: a fun design on the chaotic double pendulum. Interesting and intricate motion arises from a simple assembly of two physical pendulums (one attached to the end of the other) with the kinetic energy of the system moving back and forth between the two rings until friction damps away the energy. Chaotic motion is characterized by extreme sensitivity to initial starting conditions, tiny differences in how the system is released leads to dramatically different outcomes ea