The Norton's dome suggesting non-determinism in classical physics

Question

I am thinking of this Norton's dome.

The author guesses a solution,

$$ r(t)=\left\{ \begin{array}{c l} \frac{1}{144}(t-T)^4 & ,t\geq T\\ 0 & ,t\leq T \end{array}\right. $$

for this second order of differential equation

$$\ddot{r}=r^{\frac{1}{2}}$$

with $r(0)=0, \dot{r}(0)=0$

and then argue that classical mechanics does not have to be deterministic, given $T$ can be any number we want (as long as greater than zero).

I have a background of physics major, and understand (somewhat by this argument) that determinism as an assumption in classical physics - that we have to assume the conditions of uniqueness theorem always exist in nature for classical physics - but I would like a complete understanding of this topic, and hence I would like to ask,

Does the solution exist? Does the author's argument stands in an rigorous mathematical standing point?

Answer 1

The function is a valid solution of the differential equation. There is nothing wrong mathematically.

Whether it follows that classical mechanics is not deterministic depends on what you consider to be classical mechanics. Note that the equation of motion was derived using a constraint (that the trajectory lies on the surface of the dome), which is an idealization of the forces actually acting between the particle and the dome. If the dome itself were modelled as a classical body of finite mass, the particle could not suddenly gather momentum out of nowhere in violation of the law of conservation of momentum. Thus, for one, you have to consider idealized constraints as part of classical mechanics to reach your conclusion based on this example.

For other reflections on and possible objections to the argument from Norton's dome, see e.g. Malament, D.B., Norton's Slippery Slope, Philosophy of Science, $75$ (December $2008$), pp. $799$-$816$, and for other arguments in favour of indeterminism in classical mechanics, see e.g. the article on causal determinism in the Stanford Encyclopedia of Philosophy.

Answer 2

Norton's dome is not evidence for non-determinism. The dome curve equation admits solutions which are non-Newtonian in one singular point in the infinity of positions in state space.

It's trivial to show Norton's extra solution, whilst being a correct solution to the differential equation, is not Newtonian at the 'singularity'. This is because it has a constant value for jounce or snap, with is a 4th order term that is the "acceleration of acceleration". This is what allows the particle to move off the apex despite zero velocity and acceleration there.

However, this in itself isn't anything to do with determinism, since his equation is still deterministic, it just isn't Newtonian. (It also violates conservation of energy and momentum, which nobody seems to notice but again is trivial to show).

The reason for the apparent non-deterministic result, is that he then creates a piecewise equation, stitching the two solutions together at arbitrary time t which is plain bizarre. It has no physical justification at all! The solutions are both mathematically correct, but even if both were Newtonian, you can't just stitch equations with different initial conditions together and claim that still represents physics. It's just nonsense.

I've given a more detailed breakdown of why he's wrong in this article if you're interested: https://blog.gruffdavies.com/2017/12/24/newtonian-physics-is-deterministic-sorry-norton/

Hope that helps!