Elastic Chaos
elastic l=1 m=1 k=29.43 a=α s=0
Open in the app The dials and keys named below are the app's.
The setup
A bob of mass 1 hangs from a pivot on a spring of natural length 1 and stiffness 29.43, which would hold it at rest stretched by a third. Here it starts at its natural length and at α = 3 radians from straight down, 172°, almost straight above the pivot. It falls, the spring snaps taut and whips it round. In the run that follows the spring is between 0.18 and 2.53 long, and the bob flies from 1.18 under the floor to 1 above the pivot.
Why a spring is enough
A rigid pendulum has one angle, and with one degree of freedom its energy alone fixes the path it takes, so it cannot be chaotic however hard it swings. The spring adds a second degree of freedom, its length, and two that trade energy are enough. Start a second run a thousandth of a radian higher, and the two bobs are a whole unit apart after 5.7 seconds at Speed 1. The Double Pendulum preset, two rigid rods, parts in 4.5 seconds.
The slider
The slider α is the starting angle, and the higher the start, the faster runs part. At 2.9 the twins are a unit apart after 20.5 seconds, and at 3.14 after 2.5 seconds. Below that the parting slows sharply: at 2.7 and at 2.5 they stay less than a unit apart for a whole minute. At 2 they stay within 0.02.
Try
- Drag α to 3.14, the bob balanced upright, and the path is unpredictable after 2.5 seconds.
- Drag α to 1.5, just under the pivot's level. The spring stays between 1 and 1.98 long, and the bob swings and bounces in a steady pattern.
- Stiffen the spring tenfold, k=294.3. It barely stretches, and two runs a thousandth apart stay within 0.6 of each other for a minute: the spring is turning into a rigid rod.
- Rewind runs the identical path again. The motion is unpredictable, not random.
- Front in the deck shows the plane of the swing face on.