The Double Pendulum Model is a physical system consisting of one pendulum attached to the end of another. While a single pendulum swings in a highly predictable, rhythmic loop, a double pendulum displays chaotic, complex, and unpredictable behavior. It is one of the most famous models used in physics and mathematics to study chaos theory. System Setup
The standard mathematical model of a double pendulum consists of: Two rigid, massless rods of lengths L₁ and L₂.
Two point masses (m₁ and m₂) attached at the ends of the rods.
Two generalized coordinates representing the angular positions: θ₁ (angle of the upper rod from the vertical) and θ₂ (angle of the lower rod). The Dual Nature: Order vs. Chaos
The model splits into two entirely different behaviors depending on how much energy is in the system: 1. Low Energy (Linear Behavior)
When the pendulum is released from very small angles (θ₁, θ₂ ≈ 0), it acts like a linear coupled oscillator.
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