The double pendulum and deterministic chaos

What just happened?

Both pendulums are computed from the same equations of motion — there is no randomness here. And yet a tiny difference in the start (0.001 rad) builds up, and after a few seconds the paths of the tips have nothing in common. This is the essence of deterministic chaos.

Sensitive dependence on initial conditions:

The divergence $\delta$ of two almost identical states grows exponentially in time ($\lambda$ is the Lyapunov exponent). That is why even perfect knowledge of the equations is not enough for a long-term forecast — the error in the starting measurement explodes.

The invariant — conservation of energy:

Chaos does NOT mean “a mess with no rules”. Although the trajectories diverge, the total energy of the system stays continuously conserved — in the panel on the right the top of the bar does not wobble, only the split into Eₖ and Eₚ changes. Determinism ≠ predictability.

Where will you meet this?

The weather behaves the same way: the model is deterministic, but a small error in the initial conditions makes a 2-week forecast practically impossible. Hence the “butterfly effect”.

Why you need this for the exam:

This extends the topic of oscillatory motion and the pendulum. A simple mathematical pendulum is predictable — its period $T = 2\pi\sqrt{L/g}$ does not depend on the amplitude (for small swings). Adding a second arm destroys that regularity, but conservation of energy $E_k + E_p = \text{const}$ still holds — and that is what you use most often in problems.

Remember: a deterministic system can be unpredictable — a tiny difference in the start grows exponentially (the butterfly effect). But the total energy is always conserved: E = Eₖ + Eₚ = const.