To Move Or Not To Move...

Considering I was an utter washout at mathematics in school - for reasons too numerous to mention - it's a remarkable fact that I still find the subject one of fascination to me; even though, as I've written before, I don't always entirely grasp the detail of the stuff I read or [mostly] watch on YouTube. There is ample good content out there by numerous well qualified makers, some of whom are excellent teachers in their own right. As I've said before, one of my favourite is "Mathologer", to whose channel I'm subscribed. However, the following I discovered only today on a channel called "Up and Atom", to which I also subscribed after watching her excellent piece on something I'd not heard of before: The Dome Paradox.

This thought experiment  was arrived at in '...a long afternoon...' by John D. Norton in 2003, and has courted much controversy and debate in the intervening two decades since publication. It sets out to demonstrate that Newton's Laws of Motion do indeed permit non-deterministic systems within his otherwise fixed mechanics: that there are behaviours which cannot be predicted by The Laws. In his experiment, he pictured the above: a theoretical domed [non-hemispherical] surface having zero friction or surface deviation - no lumps or bumps or stopping power, atop which sits, at its very apex, a theoretical point mass, resting with zero force acting on it, save that of gravity, acting directly on the mass at a normal to the [mathematical] plane on which the dome rests, ie. straight down through the mass and the apex of the dome.

According to Newton's Second Law, put crudely, if something moves, it stays moving until it meets an opposing force - given the hypothetical constraint of there being zero friction. If something is not moving, and given the same theoretically reductive constraints, it stays put until it receives the input of a force. If, as in the case of the point mass on the dome, it is subject to a single constant force [gravity], with nothing to stir it from its rest, it should remain at rest forever. However, the moment a force tangential to that of that of gravity is applied, the mass will slide down the dome on any one of an infinite number of paths in the opposite direction to the force applied, like a ball off a bat [given ideal, theoretical conditions].

Now the crux of this thought-experiment is essentially that given the initial position of stasis due to the Second Law, which says if it doesn't move it will stay unmoved, balanced against the fact that the point mass is perching atop a similarly theoretical point apex having zero friction, at some point it will move in an indeterminate direction down the curvature of the dome under the influence of gravity. As if to hammer down the point, Newton's Laws also state that the principle of reversibility holds: that any Law applies in reverse; that as much as the Second Law applies to the mass rolling 'down' the dome, its principle applies in the reverse case to the mass rolling from the base to the apex and taking its position of rest, apparently with zero force applied to it, in contradiction of Newtonian mechanics whilst at the same time obeying its mathematics.

To get any further into the arguments surrounding this thought-provoking thought-experiment gets me into territory where I can't follow the maths and so there's little point in trying to go there, but I would suggest heading to "The Dome Paradox - a Loophole in Newton's Laws" on Up and Atom's YouTube channel for a very accessible explanation, which contains only a moderate dive into the maths in the process. For further detail, there's the Wikipedia entry and Norton's own pages, let alone hundreds of affirmations and counter-arguments out there in the academic literature. All fascinating stuff, including what for me is the memorable phrase that the moment in time that the mass begins to move is rather the last moment it didn't move...