[pfortuny]

> there is no equation (that we know of) that would allow us to write down where the pendulum would be at some point in the future, given its current position.

This is a misunderstanding.

The author uses the sine and cosine functions as if they were “functions which give us values” but if you are allowed to assume that (what is sin(1), by the way?), then one might as well define “Pend2(t)” as “the solution to the double pendulum equation”, and be done with it.

Which, by the way, is how one defines the exponential, trigonometric, even n-th roots! Not to say the erf, Bessel, hypergeometric functions etc.

The fact that there is not a “closed solution involving only elementary functions” is irrelevant as long as the equations ones is solving have a unique solution.

Edit: toned down.

[JackFr]

You misunderstand sensitive dependence on initial conditions, though admittedly it’s poorly elucidated in the article.

It’s not about closed form solutions. It’s about how neighborhoods on the line are mapped.

[pfortuny]

That does not matter: either the solutions are unique or not, and ODEs (out of singular points) have a unique aolution for any set of initial conditions.

The sensitivity to initial conditions has nothing to do with regular ODEs and uniqueness.

[freemint]

> ODEs (out of singular points) have a unique solution for any set of initial conditions

I am not sure if that statement is to weak. In general you can only guarantee the solution of an explicit ODE over some interval if the right hand side is Lipschitz continuous.

[pfortuny]

Well, yes. But the author is assuming it and his equations are C-infinity as a matter of fact.