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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Proof of Chaos: Measuring the Lyapunov Exponent 📈 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
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Coupled Pendulum — The Wonder of Science
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Show that, for the system of coupled pendulums shown in Fig. 2.16, the equation of motion for the n th pendulum bob is given (for small oscillations) by (d^2 ψn)/(d t^2)=-(g)/(l) ψn (a K)/(M)((ψn 1-ψn)/(a))-(a K)/(M)((ψn-ψn-1)/(a)) Show that the general solution for a mode, without regard to boundary conditions, is ψn(t)=cos(ωt φ)[A sinn k a B cosn k a] Show that the dispersion relation is ω^2=(g)/(l) (4 K)/(M) sin^2 (k a)/(2) Show that, for the boundary conditions shown in Fig. 2.16 (i.e., no s
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