I find this write-up enlightening : https://blog.gruffdavies.com/2017/12/24/newtonian-physics-is...

It seems the equations given by Norton really are unphysical, so actually it's not a proof on indeterminism, but simply incompleteness - newtonian physics cannot model some really really fucky situations - its incomplete.

I can recommend taking this blog post with a grain of salt. I'm a physics masters student and after working through the math myself I believe the Lipschitz continuity violation that Gruff rejects as a red herring is actually the real source of the nondeterminism, and is not just some mathematical fluff.

The first law and stitching arguments he makes appear to both be flawed. Having non-zero derivatives of force in combination with zero velocity and zero force is perfectly in accordance with Newton's first law. And in his frictionless ball counterexample, his equation is incorrect because it violates Newton's second law, not because two solutions are stitched together.

Lipschitz continuity is required for guaranteed uniqueness of differential equation solutions, and non-uniqueness can appear as nondeterminism or incompleteness.

I think he reaches the right conclusion but his reasoning is flawed.

That’s a brilliant analysis of the Norton’s dome; thank you.

Thanks. It's a great piece of writing!

Why is virtual work not taught anymore?

Who says it’s not? I suspect most courses in advanced classical mechanics still cover it, although probably briefly.

Norton’s dome is intriguing; other departures from determinism in classical physics include “space invaders”: in some many-particle systems a particle can develop an infinite velocity, which sends it “out of world”; then time reversibility means it can enter the world unpredictably. I mention both of these in an article about unsolved problems in classical physics: http://arstechnica.com/science/2014/08/the-never-ending-conu...

Doesn't conservation of energy preclude this? If all particles start with finite energy, and only finite energy is added to the system, won't the total energy stay finite? Or are you assuming an infinite potential somewhere, e.g. gravity from a point "planet"?

The examples I know of all involve gravitational or similar potentials, so there is unbounded negative energy available from 1/r. But you can still get singularities without collisions: Xia, Annals of Math. 135 411–468 (1992).

At least a few of these seem to be cases where the concept of a point particle breaks down.

There are similar problems in E&M at caustics that are solved by remembering that E&M waves have non-zero wavelength.

>which sends it “out of world”

Wouldn't any system have to include this "out of world" particle? Doesn't its inclusion mean "the world" simply grows at that infinite rate?

> Take a simple bottle of milk like the ones pictured here, with a cylindrical section below a tapering segment.

Those bottles are rectangular!

They are rectangular cylinders. The shape works for the problem under discussion.

I thought you where making up words, saying “rectangular cylinder”, but apparently it’s a thing:

https://www.quora.com/Geometry-What-is-a-rectangular-cylinde...

A cylinder can have any cross section.

Norton doesn't understand the principle of virtual work and should go back and read D'Alembert and Lanczos.

Link to actual explanation from Norton: http://www.pitt.edu/~jdnorton/Goodies/Dome/

Are these kinds of math tricks less likely to be a lack of modeling than nondeterminism, ie, does the fact that one mathematical model produces a non-deterministic result mean that the actual situation is non-deterministic?

No, it doesn’t, as you suspect. The article that sudosysgen links to up above treats this question nicely.