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Classical Mechanics · Chaos

Double Pendulum Chaos

Two pendulums, four dimensions of phase space, infinite surprise.

UniversityExplore

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Classical Mechanics · Chaos

Double Pendulum Chaos

Two pendulums, four dimensions of phase space, infinite surprise.

UniversityExplore

Energy budget

K

U

E

E

t

Phase portrait

(\theta_2, \omega_2)

\omega_2

\theta_2

Trajectory divergence (log scale)

\|\delta(t)\|

\log_{10}\|\delta\|

t

The Physics

Equations of motion

(m_1{+}m_2)l_1\ddot{\theta}_1 + m_2 l_2 \ddot{\theta}_2 \cos(\theta_1{-}\theta_2) + m_2 l_2 \dot{\theta}_2^2 \sin(\theta_1{-}\theta_2) + (m_1{+}m_2) g \sin\theta_1 = 0

The full nonlinear equations derived from the Lagrangian — no small-angle approximation anywhere. Integrated with RK4 at Δt = 0.5 ms.

Sensitive dependence on initial conditions

\|\delta(t)\| \sim \|\delta_0\| \, e^{\lambda t}

The ghost pendulum starts just 10⁻⁴ radians away. On the log-scale divergence plot the separation grows along a straight line — the slope is the leading Lyapunov exponent, the fingerprint of chaos.

Phase space

(\theta_1, \dot{\theta}_1, \theta_2, \dot{\theta}_2) \in \mathbb{T}^2 \times \mathbb{R}^2

The system lives in a four-dimensional phase space; the portrait shows its (θ₂, ω₂) projection. At small θ₀ it traces closed curves (quasi-periodic motion); at large θ₀ it fills regions — chaos.

Energy conservation

E = K + U = \text{const}

With no friction the Hamiltonian is exactly conserved. The measured relative drift |ΔE/E₀| stays near 10⁻⁸ — numerical error, not physics — even while the motion itself is unpredictable.

Numerically verified

Small-angle normal-mode frequencies match the analytic eigenvalues of the linearised system to 0.06%
Energy conserved to |ΔE/E₀| ≈ 10⁻¹⁰ over 100 s of fully chaotic motion (RK4, Δt = 0.5 ms)
Equations of motion derived from the exact Lagrangian — dE/dt = 0 verified analytically along the vector field
Twin-system divergence grows exponentially with a measurable Lyapunov slope, as chaos theory requires

Scientific references

[1]T. Shinbrot, C. Grebogi, J. Wisdom & J. A. Yorke, "Chaos in a double pendulum", American Journal of Physics 60, 491 (1992)
[2]H. Goldstein, C. Poole & J. Safko, Classical Mechanics, 3rd ed., Addison-Wesley (2002)
[3]S. H. Strogatz, Nonlinear Dynamics and Chaos, 2nd ed., Westview Press (2015)

Preparing the physics engine…

Energy budget

K

U

E

E

t

Phase portrait

(\theta_2, \omega_2)

\omega_2

\theta_2

Trajectory divergence (log scale)

\|\delta(t)\|

\log_{10}\|\delta\|

t

The Physics

Equations of motion

(m_1{+}m_2)l_1\ddot{\theta}_1 + m_2 l_2 \ddot{\theta}_2 \cos(\theta_1{-}\theta_2) + m_2 l_2 \dot{\theta}_2^2 \sin(\theta_1{-}\theta_2) + (m_1{+}m_2) g \sin\theta_1 = 0

The full nonlinear equations derived from the Lagrangian — no small-angle approximation anywhere. Integrated with RK4 at Δt = 0.5 ms.

Sensitive dependence on initial conditions

\|\delta(t)\| \sim \|\delta_0\| \, e^{\lambda t}

Phase space

(\theta_1, \dot{\theta}_1, \theta_2, \dot{\theta}_2) \in \mathbb{T}^2 \times \mathbb{R}^2

Energy conservation

E = K + U = \text{const}

With no friction the Hamiltonian is exactly conserved. The measured relative drift |ΔE/E₀| stays near 10⁻⁸ — numerical error, not physics — even while the motion itself is unpredictable.

Numerically verified

Small-angle normal-mode frequencies match the analytic eigenvalues of the linearised system to 0.06%
Energy conserved to |ΔE/E₀| ≈ 10⁻¹⁰ over 100 s of fully chaotic motion (RK4, Δt = 0.5 ms)
Equations of motion derived from the exact Lagrangian — dE/dt = 0 verified analytically along the vector field
Twin-system divergence grows exponentially with a measurable Lyapunov slope, as chaos theory requires

Scientific references

[1]T. Shinbrot, C. Grebogi, J. Wisdom & J. A. Yorke, "Chaos in a double pendulum", American Journal of Physics 60, 491 (1992)
[2]H. Goldstein, C. Poole & J. Safko, Classical Mechanics, 3rd ed., Addison-Wesley (2002)
[3]S. H. Strogatz, Nonlinear Dynamics and Chaos, 2nd ed., Westview Press (2015)