Determinism and Norton’s Dome

1. The curve is cut off at the upper end (once).
2. I read between the lines of your answer that you find it intuitively difficult to accept a theory that allows for non-determinism in certain extreme cases.
Is this true? And if so, why exactly? Maybe this is a “good” property of the theory?

The question is whether a form that is not derivative from second order onwards exists in nature. My feeling is that it does not.
Indeed it is difficult, because the laws are supposedly deterministic. The fact is that they present it as a paradox. Something is happening here for no reason, and this is against the principle of causality.

It would be nice if the rabbi wrote a column about this sometime, because it really sounds amazing, but as far as I'm concerned, the above entry is part of the Chinese Wikipedia.

1. Regarding shapes in nature – I think it depends on how much zoom-in you do. At the atomic level, it seems to me that there are no deductions even once (and maybe even no continuity).

2. Norton explains that there is no paradox if you formulate the first law like this: “If the sum of the forces is zero – the acceleration will be zero”: By the time the body starts moving (t<T), its acceleration is indeed zero (and its velocity is also zero). At the moment t=T its acceleration is still zero (because of the shape of the curve). At T<t there is acceleration, but the sum of the forces is no longer zero. There is no point in time when the law is violated, so there is no paradox.
https://sites.pitt.edu/~jdnorton/Goodies/Dome/

3. The analysis does not show that something happens without a (physical) reason, but that in this system, at any moment something like this can happen. In my opinion (and my hope), perhaps this opens up the possibility of events whose cause is not physical and which do not violate the laws of physics. In other words: perhaps the laws of nature are willing to sometimes allow additional factors to influence reality.

4. Note: Not every convex curve can produce this phenomenon. If we take a spherical dome for example, the time it takes for a mass to gain speed is infinite, and therefore the motion cannot actually begin. Norton's innovation is in creating a curve in which time is finite.

1. When you are at the atomic level, you have to use quantum theory and the whole calculation is irrelevant.
2. I didn't understand. At the moment the movement begins, there is still no force. This is probably an expression of the lack of determinism there at the tip.
3. I didn't understand the difference. The problem is that something like this can happen.
4. It's clear. The wiki argued the other way around. If you look at a ball that climbs up and stops at the tip, it will take infinite time.

I have now uploaded (column 687): https://mikyab.net/wp-content/uploads/2025/01/%D7%98%D7%95%D7%A8687-%D7%A4%D7%A8%D7%93%D7%95%D7%A7%D7%A1-%D7%A0%D7%95%D7%A8%D7%98%D7%95%D7%9F.pdf