The Dome Paradox: A Loophole in Newton's Laws

There are many good treatments of this supposed loophole. I happen to like this one:

https://blog.gruffdavies.com/2017/12/24/newtonian-physics-is...

It points out many flaws in Norton's reasoning, some fatal to his argument, some not. Putting it as simply as I can, Norton seems to claim that "Newton's Laws" are non-deterministic. That's not quite right. Rather, they are non-complete. I.e. they are incomplete. They're incomplete insofar as Newton's First Law ("An object at rest remains at rest, and an object in motion remains in motion at constant speed and in a straight line unless acted on by an unbalanced force") establishes first-order and second-order derivatives (momentum and acceleration) as state variables but places no constraints on higher-order derivatives. However, higher-order derivatives are (as many as are needed) among a system's state variables. In many (but far from all), higher-order derivatives are zero and human experience with them is rare, so they're easy to overlook. Norton's (unphysical) Dome is a specific example of a general class of systems where higher-order derivatives are not zero. Given that, the two branches of Norton's equation of motion (for the stable and unstable trajectories) cannot both describe the same system (or the same particle) with the same set of state variables. That's the sleight-of-hand.

Again, all credit to Gareth Davies for working this out. I am absolutely not trying to pass off his work for my own. Just reporting it and trying to summarize it.

Related ongoing thread:

The Dome (2005) - https://news.ycombinator.com/item?id=42583688 - Jan 2025 (22 comments)

Is there a way to get the key points of this interesting topic without having to watch a 23 minute video?