[jdhwosnhw]

Norton’s dome is a valid paradox, in the sense that the math really does admit two valid equations of motion. The link you provided doesn’t dispute that fact (other than commenters pointing out that you need a proportionality constant to make the units work out). My favorite intuitive explanation for the presence of the paradox is well summarized by the Wikipedia article on the dome: “To see that all these equations of motion are physically possible solutions, it's helpful to use the time reversibility of Newtonian mechanics. It is possible to roll a ball up the dome in such a way that it reaches the apex in finite time and with zero energy, and stops there. By time-reversal, it is a valid solution for the ball to rest at the top for a while and then roll down in any one direction.”

[Retric]

Calling it a valid paradox is questionable, there’s a little mathematical sleight of hand required for the particle to actually stop in finite time. It doesn’t work for say particle sliding up a sphere.

To work the curvature of the dome is infinite is at the apex, which then breaks many things. There’s a lot of disagreements around this paradox and much older related examples because Newtonian physics is somewhat ill defined: https://philsci-archive.pitt.edu/8833/1/dome_v3.pdf

[ASalazarMX]

Since many thought experiments allow for objects to be composed of infinite one-dimensional points that somehow form higher-dimensional bodies, an infinite curvature could be interpreted as the apex being a single point supporting the point at the bottom of the ball. Both points should be perfectly aligned and in perfect equilibrium.

It has no reason to roll unless the placement was uneven, and if it was uneven, it would not break determinism.

[pxx]

I don't know if you can actually stitch the equations together though because they have different initial values, albeit in higher order derivatives than Newtonian mechanics cares about.

see https://blog.gruffdavies.com/2017/12/24/newtonian-physics-is... which was linked to from that new question:

> If we start at an arc length of 1/144 for example, it will run up the dome and arrive at the apex in 1 second. As we’ve seen, it has zero velocity and zero acceleration at this point, but moves off after anyway because it still has a positive value for snap.

[adrian_b]

Norton's dome is not a valid paradox, it just exploits in an overly complicated way the fact that almost no handbook of physics bothers to present a complete set of axioms for the classical mechanics (and even less for relativistic or quantum mechanics). So the "paradox" is based on the sloppy teaching of physics from a mathematical point of view.

The form of the Norton's dome does not matter. The so-called paradox is just a random example of the fact that there exist multiple functions of a variable that have in the origin the same values for the function and for the first 2 derivatives, e.g. various pairs of polynomials of the 4th order.

Therefore if you accept any function of time as describing a possible motion, you can always find motions that at some moment in time have the same position, velocity and acceleration.

This is not an example of indeterminacy in classic mechanics, because one of the axioms of the classic mechanics is that all the forces that exist in nature are such that the state of a mechanical system is completely determined by the positions, velocities and accelerations of its components (in other words, a mechanical system must be described by a system of differential equations of the second degree that has a unique solution).

There is no difficulty of imagining other kinds of forces, for which this assumption is not true, but a theory where such forces exist is no longer the Newtonian mechanics, in the same way as any geometry where Euclid's axiom of parallels is not true is no longer an Euclidean geometry.

If Newtonian mechanics were a correct model for the World, a ball would remain forever on the top of the dome, without ever falling. In reality, even assuming the validity of Newtonian mechanics, the main reason why any attempt to test this experimentally would fail is the thermal motion, due to which a ball can never be at rest, so it would always start immediately to fall in a random direction.

The violation of the axioms is why the so-called different solutions are not solutions within Newtonian mechanics.

On the other hand the argument that the initial state could be obtained by launching the ball towards the top, and then time reversal would demonstrate a valid solution, it is also wrong, because the so-called solution cannot be obtained by time reversal.

If the ball is launched with only enough energy to reach the top, so it will come to rest, then it requires an infinite time to reach the top. Reversing the time means that the ball will remain on the top for an infinite time, without falling, as expected.

[ASalazarMX]

The real paradox is why and how a stationary object in a perfect world that obeys Newton's laws suddenly started moving. There's only mass and gravity, not even thermal or atomic effects allowed. The best justification the author gives is that it happened and this thought experiment doesn't explain why, how, or even when.

TL;DR: Magic breaks Newton's laws