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Determinism

Two pendulums, identical to a thousandth of a radian, ending up in different lives.

Two double pendulums, released from angles that agree to a thousandth of a radian. For a few seconds they move as one. Then they don't. Within ten swings they are doing entirely different things, and nothing went wrong. The equations are exact, deterministic, and utterly unpredictable.

The concept

Determinism does not mean predictability. A double pendulum is fully governed by Newton's laws, no randomness anywhere, yet it is chaotic: the tiniest difference in starting conditions grows exponentially until the two histories share nothing. The future is fixed by the present, and the present can never be known precisely enough to compute it for long.

The math

Nearby trajectories separate at a rate set by the largest Lyapunov exponent $\lambda > 0$:

δ(t)≈δ0 eλt\delta(t) \approx \delta_0\, e^{\lambda t}

An initial gap $\delta_0$ blows up to order-one in a time $\sim \frac{1}{\lambda}\ln(1/\delta_0)$. Because the dependence on $\delta_0$ is only logarithmic, buying a thousand times more precision buys you just a little more predictable time. The pendulum's own motion follows the coupled Euler-Lagrange equations for the two angles, integrated with a small enough step that the chaos is physical, not numerical.

Why it stays strange

This is why weather forecasts die at about two weeks no matter how good the satellites get: the atmosphere is a double pendulum with billions of joints. Laplace imagined a demon who, knowing every position and velocity, could compute all of history. Chaos does not refute the demon. It just proves that anything short of infinite precision is, in finite time, blind.

Further reading

  • Lorenz (1963), "Deterministic Nonperiodic Flow" - chaos's founding paper.
  • Strogatz, Nonlinear Dynamics and Chaos.
@article{lorenz1963,
  author  = {Lorenz, Edward N.},
  title   = {Deterministic Nonperiodic Flow},
  journal = {Journal of the Atmospheric Sciences},
  volume  = {20},
  number  = {2},
  pages   = {130--141},
  year    = {1963}
}
@book{strogatz2015nonlinear,
  author    = {Strogatz, Steven H.},
  title     = {Nonlinear Dynamics and Chaos},
  edition   = {2nd},
  publisher = {Westview Press},
  year      = {2015}
}