Here's how I think about it:

For any dome-like shape, you can start a marble at the bottom and roll it up with some initial speed. If you roll it with insufficient initial speed it'll turn around and come back down. If you roll it too hard, it'll overshoot the peak. By continuity, there must exist some initial condition where it stops at the top.

Now, here's the thing that makes Norton's dome special: For a typical dome shape it'll take an infinite amount of time before that marble stops at the top. If you plot the position as a function of time it'll have some type of sigmoid-like shape. However, for the special case of Norton's dome, you can make it settle at the top in a finite amount of time where it'll sit for the rest of eternity. In other words, if you plot the position as a function of time, there will be some critical time after which its position is constant.

Now, the clever thing to do now is to realize that Newton's laws are time reversal symmetric which means that any motion forward in time could equally well happen backwards in time.

So, you're allowed to take any position plot and flip it horizontally; this is also going to be a valid trajectory.

For any typical dome shape this is not a problem. For a typical dome shape you have a sigmoid-like solution which, when flipped, is still sigmoid shaped. In particular this means that there is no finite time at which you can place the marble at the top of the dome and have it roll off. At any finite time, the marble will be slightly off the top and have a small nonzero speed.

Norton's dome is different. If you flip its trajectory horizontally you'll see that there are many moments in time where you can start the marble at the top to have it abruptly start rolling off the top at some later time. This is the paradox. You can choose to have it sit at the top for one second and then start rolling or sit at the top for one minute and then start rolling.

Unlike other domes, Norton's dome seems to violate our intuition for how initial conditions work. In all cases the marble starts at the top with zero initial speed and yet falls off the top att different moments.

The answer given in Stack Exchange and the Wikipedia article linked to in one of its comments provide explanations, though whether you find them convincing is, of course, up to you!

The Wikipedia article says there are solutions to the classical equations of motion: one in which the ball remains stationary forever, and then all those where, after an arbitrary period during which the ball is stationary, it rolls off the dome in an arbitrary direction. What makes this indeterminate is that the analysis of a single initial state yields multiple possible outcomes.

The article goes on to say "Notice in the second case that the particle appears to begin moving without cause and without any radial force being exerted on it by any other entity, apparently contrary to both physical intuition and normal intuitive concepts of cause and effect, yet the motion is still entirely consistent with the mathematics of Newton's laws of motion so cannot be ruled out as non-physical."

This raises the question of what we mean by 'physical', and whether theories of physics define the physical or describe something existing independently. I will leave that to the more philosophically minded; for myself, I will just note that as there are cases (and physically realizable ones at that) where classical physics gives answers that are not merely indeterminate but outright wrong (the ultraviolet catastrophe being a canonical example), I don't think anything of consequence hangs on this particular case.

https://en.wikipedia.org/wiki/Norton%27s_dome#Solutions_to_t...

I looked over the explanation on the Wiki and, thinking about it, I'd say that the surface around the tip of Norton's Dome is not a continuous function, which alone is enough for me to disqualify it.

Intuitively, that's a sufficient explanation to me, or at least a sufficient start of one. IANAPhysicist, so I'll ask here: are there any examples of surfaces or phenomena in classical physics that are defined by a discontinuous function, and are something you'd actually expect to see existing in the real world? Things seemingly discontinuous until you zoom in close enough don't qualify.

Pontus gave a good explanation of the physics. Mathematically the statement is simply that there are different trajectories with the same initial conditions that solve the same differential equation. This is easy to calculate and has nothing to do with "people wanting it". It's simply a mathematical fact. What do you not find convincing here?

This is the first time I'm encountering Norton's dome, and I'm not particularly academic, just really like learning about math -- so I'm hoping a friendly HNer can help me out here.

Is norton's dome essentially describing a saddle point? Is the only reason it's nondeterministic because at that point things go to infinity? If we're in the world of mechanics, wouldn't it be up to the machine to determine what to do at that point? Implementation defined, one might say?