>[!abstract] >Norton's dome is a [[Gedanken|thought experiment]] that exhibits a non-deterministic system within the bounds of Newtonian mechanics. It was devised by [[Norton, 2008|John D. Norton]] in 2003. It is a special limiting case of a more general class of examples from 1997 by Sanjay Bhat and Dennis Bernstein. The Norton's dome problem can be regarded as a problem in physics, mathematics, and philosophy (Wikipedia, 2025). >[!abstract] Thought experiment >Assume a ball sitting on the apex of a dome described by the following initial values: >$r'' = r1/2$ >$r(0) = 0, r'(0) = 0$ >This problem has two solutions because the Lipschitz condition is not met in the neighborhood of $r=0$. One of these solutions is non-zero and seems to suggest that the ball will start moving on its own at some point, in apparent violation of the law of conservation of momentum, as if the universe was non-deterministic. >[!note] >I understand that Norton's Dome is generally taken by mathematicians and physicists to **not** be evidence of non-determinism, despite the original paper's claim. What this [[Gedanken]] (and, importantly, not a physical experiment) shows is that, in this special configuration of a dome, the Newtonian mechanics equations have more than one solution, and we do not know how to interpret the existence of multiple solutions. However, if we force a [[Lipschitz condition]] (of existence and uniqueness), we can view Norton's dome as a mathematical object that has no corresponding physical ontology, and the conundrum disappears. This is analogous to mathematical results leading to infinities despite the absence of singularities in the physical universe. Norton's dome is not smooth (it has infinite curvature at its apex), and if it was made to be smooth, the initial-value problem would be Lipschitz and have only one solution, as expected in Newtonian mechanics. >[!references] >- [[Norton, 2008]] for the original Gedanken >- [[Fletcher, 2012]] for a discussion of what constitutes a Newtonian system >- [Reddit thread](https://www.reddit.com/r/askscience/comments/6kxto3/comment/djpojxq/) >- [Causal determinism](https://plato.stanford.edu/entries/determinism-causal/#ClaMec) on the Stanford Encyclopedia of Philosophy >[!related] >- **North** (upstream): [[Newtonian mechanics]] >- **West** (similar): [[Boussinesq’s problem]] >- **East** (different): [[Determinism]] >- **South** (downstream): [[Non-determinism]]