Computational Study of Chaos and Sensitivity to Initial Conditions in a Frictionless Double Pendulum System

This paper presents a rigorous computational study exploring deterministic chaos and sensitivity to initial conditions within a frictionless double pendulum system. Using Lagrangian mechanics, the system's coupled, non-linear second-order differential equations of motion are derived and subsequently transformed into a state-space formulation. Numerical integration is performed using the Runge-Kutta 45 (RK45) algorithm with a strict time step (\(dt = 0.01\)) to actively mitigate and control the inherent energy drift associated with non-symplectic integrators. By simulating high-energy configurations (\(θ_1 = θ_2 = 90^∘\)) and introducing a microscopic perturbation (\(Δθ_2\)) to the initial state, the study demonstrates and quantifies the exponential divergence of phase space trajectories. The outcomes effectively illustrate the limits of long-term predictability in classical chaotic systems. Full source code for the simulation and data analysis is provided via the linked repository for exact reproducibility.