I read the paper Example of indeterminacy in classical dynamics. I understand the paper: Because the differential equation does not satisfy the Lipschitz condition, its solution is not unique, and thus there is indeterminacy.
But is it really possible in physics, as in the example described in the paper, for the ball to rest at the origin to suddenly deviate to the left or right from the origin, which violates the law of causality? Or does the physical world not follow the law of causality?
We can’t possibly know.
We know that classical mechanics completely breaks down on atomic scales (in fact, classical mechanics can be considered just a simplifying limit case of quantum mechanics). However, if we impose a lower scale limit that we can reasonably consider, every surface fulfils the Lipschitz condition.
On top, any experiment testing this would require us to create a perfectly round ball and perfectly place it on a perfectly crafted surface in a perfect vacuum. This is clearly impossible in our world due to quantum effects, the atomic nature of matter, etc. (Of course, you can consider an analogous experiment that doesn’t require a ball and such, but you would run into similar issues.) We can’t say how things would be in a hypothetical universe that is classical all the way down – because we are not living in that universe.