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Classical Mechanics · Chaos

Double Pendulum Chaos

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

Preparing the physics engine…

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Classical Mechanics · Chaos

Double Pendulum Chaos

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

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

From the Lagrangian

L = T - U, \qquad \frac{d}{dt}\!\left(\frac{\partial L}{\partial \dot{\theta}_i}\right) - \frac{\partial L}{\partial \theta_i} = 0

Rather than chase the tension forces in each rod, we write one scalar — the Lagrangian L = kinetic minus potential energy in terms of the two angles — and turn the Euler–Lagrange machinery. It delivers the equations of motion automatically, the elegance of analytical mechanics: geometry of energy in, dynamics out.

The coupled 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 result is two equations locked together through the cos(θ₁−θ₂) coupling — fully nonlinear, with no small-angle approximation anywhere. That nonlinearity is precisely what makes the motion so rich. They are integrated here with fourth-order Runge–Kutta at Δt = 0.5 ms, small enough that the numerical error sits far below the physics.

Sensitive dependence & the Lyapunov exponent

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

The ghost pendulum starts just 10⁻⁴ radians away — an invisible difference. On the log-scale divergence plot their separation grows along a straight line whose slope is the leading Lyapunov exponent λ, the quantitative fingerprint of chaos. A positive λ means prediction has a horizon: every extra digit of accuracy in the starting angle buys only a fixed extra span of foresight. This is the butterfly effect, made measurable.

Phase space, tori and the KAM picture

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

The state lives in a four-dimensional phase space; the portrait shows its (θ₂, ω₂) slice. At low energy the trajectory winds around invariant tori and traces closed loops — regular, quasi-periodic motion protected by the KAM theorem. Raise the energy and those tori break up, the orbit begins to fill whole regions, and the motion turns chaotic. The same apparatus is integrable in one regime and chaotic in another.

Energy is still exactly conserved

E = T + U = \text{const}, \qquad \frac{dE}{dt}=0

Chaos is not sloppiness. With no friction the Hamiltonian is conserved exactly, and the measured relative drift |ΔE/E₀| stays near 10⁻⁸ — that is numerical round-off, not physics — even while the motion itself is utterly unpredictable. Determinism and unpredictability coexist: the future is fixed by the present, yet no finite-precision measurement can pin it down.

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

From the Lagrangian

L = T - U, \qquad \frac{d}{dt}\!\left(\frac{\partial L}{\partial \dot{\theta}_i}\right) - \frac{\partial L}{\partial \theta_i} = 0

The coupled 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

Sensitive dependence & the Lyapunov exponent

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

Phase space, tori and the KAM picture

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

Energy is still exactly conserved

E = T + U = \text{const}, \qquad \frac{dE}{dt}=0

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)