A guide to reading a double pendulum

What a double pendulum is

A double pendulum is one rigid arm hanging from a pivot, with a second arm hanging from the free end of the first. Each arm carries a bob at its end. Two angles, each measured from straight down, fix the position of the whole device. Two more numbers, the rates at which those angles change, fix the motion. That is four numbers in all.

In the idealised model on the home page, both arms have the same length and both bobs have the same mass. There is no friction and no air. These simplifications keep the mathematics clean and leave the behaviour intact, which is why the model is useful for teaching.

Why small differences grow

The equations of motion are deterministic. Given an exact starting state, the future is fixed. The trouble is that no measurement is exact. Each angle has some uncertainty, and the motion carries that uncertainty forward into a different future.

The mechanism is stretching. Two nearby starts follow similar paths for a while, and the distance between them changes at a rate set by the local behaviour of the system. In a double pendulum the coupling between the arms can stretch that distance quickly for some starts and slowly for others. Some pairs of starts stay close for many swings. Others part within a few.

Two things matter. The first is how far apart the starts are. The second is how long you wait. Over a short time almost any system looks predictable. Over a long time a chaotic system does not, even though every step of it is governed by an exact rule.

Reading the twin

Turn on the twin on the home page. The twin starts from the same state as the main pendulum, except that its lower arm is moved 0.001 radians away. The gap readout gives the distance between the two lower bobs, in arm lengths. A gap of 0.0010 means the twin began one thousandth of an arm length from the main pendulum.

Usually the gap passes through three stages. At first it grows slowly, close to the starting offset. In the middle stage it grows quickly, and the two pendulums visibly disagree. In later stages the gap wanders. Two arms of fixed length cannot pull apart without limit, so the gap is bounded by their reach.

Watch for the moment the two paths stop looking like copies of each other. That moment is not set by the instrument. It depends on the starting angles, so the same offset can give a long agreement from one start and an early split from another.

What reset does, and what it does not

Reset places both arms at the angles on the sliders, sets the clock to zero and clears the trails. It does not change the physics. Two runs from the same starting angles, with the same zero starting speeds, will match. Reset is therefore a way to test a claim, not a way to make a surprise disappear.

Reset also cannot give you a true starting state from the real world. The sliders move in whole degrees, and any real measurement has uncertainty finer than that. The instrument shows the consequence of a start. It does not show the start you actually have.

Where prediction stops

The idealised double pendulum has no friction, no air and no noise. A physical device has all three, and they change the picture. Damping removes energy, so a real double pendulum eventually comes to rest, and the long-run question changes from how it moves to where it settles.

The model is still useful for the first few swings and for understanding structure. A forecast for a specific device would need measurements of its arms, bearings and starting state that most people never make. Without them, the honest answer to a long-range question is that the prediction is not reliable beyond some horizon.

The useful question is rarely whether a system is predictable in general. It is how long you can trust a prediction before the uncertainty you accept grows larger than the answer you want. Ask that question about any system you care about.

Continue with the four studies, or return to the instrument on the home page.