Chaos Theory and the Prolate Spheroid

[Ben Walker]

Rugby balls and American footballs are prolate spheroids. As such, their bounce patterns seem sporadic - they tend to bounce to different heights and in different directions even when they appear to hit the ground with a constant angle, speed, and spin. Does this behaviour relate to chaos theory, whereby the outcome is highly dependent on the initial position?

 

[jedishrfu]

Welcome to PF!

There's a book that gets into the connection called Newton's Football:

https://www.amazon.com/dp/0345545141/?tag=pfamazon01-20

and I found this excerpt:

https://books.google.com/books?id=kJep4m0DozIC&pg=PT5&lpg=PT5&dq=chaos+and+prolate+spheroid+bounce&source=bl&ots=nsaVp3tKFU&sig=_P3MsRGZNrZOV4Bn8QSQxDLbltw&hl=en&sa=X&ei=PYodVaunGYyxsAW-xILYDg&ved=0CEQQ6AEwBQ#v=onepage&q=chaos and prolate spheroid bounce&f=false

 

[Pythagorean]

To call the system chaotic, you would have to simulate it and measure a positive Lyapunov constant.

My instincts say yes, but a football on a flat field would probably exhibit transient chaos, not asymptotic chaos. That is, after the chaotic excursion, the ball would come to a non-chaotic rest state.

 

[Blackberg]

How does this compare to dropping a dice? Is there a distinction?

 

[Pythagorean]

How does this compare to dropping a dice? Is there a distinction?

Just the shape of the "ball" and various friction coefficients. Dice throw models yield positive lyapunov constant [1] (and the models can predict dice throw statistics)

[1] http://scitation.aip.org/content/aip/journal/chaos/22/4/10.1063/1.4746038

 

[Ben Walker]

Thanks a lot for the response :) How would I measure a positive Lyapunov constant? And could I do this with an actual football?

 

[Pythagorean]

Thanks a lot for the response :) How would I measure a positive Lyapunov constant? And could I do this with an actual football?

Chaos is sensitivity to initial conditions and we can't perfectly set the initial conditions of a football (or measure it's state with precision). Further, we can't guarantee elimination of any external perturbations. This is one of the interesting aspects of chaos theory - we can have systems that are deterministic, but still unpredictable.

So instead, you would have to develop a deterministic model of a football, start it with a set of initial conditions, then start another simulation with slightly different initial conditions. After the system evolves some, you would measure how much the perturbed system diverges from the unperturbed system. Then you throw away the perturbed system and take the nominal system at it's new state and perturb it, then measure divergence between the two systems. And so on. Each new measurement is averaged into the older measurements and, as time goes on, you see the Lyapunov measurement begin to approach a constant.

This is, of course, a simplification. In reality, the way you perturb the system matters (for example, perturbing one variable in the system vs. another).

 

[Ben Walker]

So there is no way I could achieve a small enough perturbation through any practical method?