In that case it is impossible for the ball to have rolled up the dome, because there is nothing in the system that could have given it the necessary upward impulse. So if we observe it sitting at the top of the dome, the only possible history is that it has always been there. This can be derived from the 2nd law alone. The 1st law is not needed.I am assuming that the system consists in the dome, the ball bearing, the ambient gravitational field, and nothing else. — Pierre-Normand
In that case it is impossible for the ball to roll up the dome, because there is nothing to give it the necessary upward impulse. So if we observe it sitting at the top of the dome, the only possible history is that it has always been there. This can all be derived from the 2nd law alone. The 1st law is not needed. — andrewk
In that case the path that involves the ball having always been at the top of the dome will not be consistent, under the 2nd law, with the current state of the cannon or the cue stick (eg heat, momentum) Also, the momentum of the dome will be different in both cases, as the ball transfers its horizontal momentum to the dome (3rd law) as it climbs to the top.Les us assume that the ball has been shot up with a canon, or hit with a cue stick, if you like. — Pierre-Normand
In that case the path that involves the ball having always been at the top of the dome will not be consistent, under the 2nd law, with the current state of the cannon or the cue stick (eg heat, momentum) Also, the momentum of the dome will be different in both cases, as the ball transfers its horizontal momentum to the dome (3rd law) as it climbs to the top. — andrewk
Gravity.
— creativesoul
What about it? — Pierre-Normand
I suppose I just got caught up in the momentum, and the challenge of coming up with arguments against a well-directed set of challenges — andrewk
Where's it being accounted for here? — creativesoul
Newtonian gravity then... — creativesoul
I suppose my simple mind is struggling to see the relevant difference between being pushed or falling...
I mean, when taking gravity into consideration... — creativesoul
Is it? Gravity is never zero. Accompanied by a significant enough amount of molecular decay of either the bearing or the dome, and it will fall...
Right? — creativesoul
The radial component of the reaction force is proportional to the sine of the slope at the point of contact with the ball, and hence null when the ball is located at the apex. — Pierre-Normand
Doesn't the net force change alongside with molecular decay? — creativesoul
Not sure what molecular decay is. But if you're thinking of thermal molecular motion, yes. It would be a source of fluctuation of the net force, and then could be appealed to as the cause of the fall. But that doesn't solve the conceptual issue since, according to Newton's laws of motion, the "fall" (or initiation of the movement) is possible even if there is no initial perturbation at all. — Pierre-Normand
I'm not seeing the need for an initial perturbation either. The system of molecular decay can change the net force causing the bearing to begin being in motion all the while never appealing to a force outside the system, aside from gravity. The physical structure of molecules changes over time. This change alone is enough to account for the movement of the bearing after sufficient time without introducing another force. — creativesoul
...it still doesn't address the initial problem regarding Newton's laws: namely, that they allow the ball to start moving even in the case where there is no such initial departure from symmetry. — Pierre-Normand
And this is solely as a result of the shape of the dome? — creativesoul
Yes. Although Norton's dome isn't the only shape that allows this, many shapes, such as a spherical dome, or a paraboloid, wouldn't allow it since it would take an infinite amount of time for a perfectly balanced ball to "fall off" from the apex. (Or, equivalently, in a time-reversed scenario, it would take an infinite amount of time for a ball sent sliding up to come to rest at the apex). — Pierre-Normand
It may be that the solution lies in what's being neglected by the problem itself — creativesoul
I have come to the tentative realisation that I don't have any intuitive sense that a model of the world needs to be satisfy that criterion. Key contributors to this feeling are:Notice, though, that this proposed expansion only shaves off 'branching outs' from bifurcation point towards the future. Determinism is commonly defined as a property of a system whereby the state of this system at a time, in conjunction with the dynamical laws governing its evolution, uniquely determine its state at any other time (either past or future from this point in time). This is a time-symmetrical definition of determinism. Under that definition, if the laws are such that there remains bifurcation points in phase space that are branching out towards the past, then the system still is indeterministic. The system past or present states uniquely determine its future; but its future or present states don't always uniquely determine its past. — Pierre-Normand
This is not just an artificial construction. It is my best attempt at stating formally what I believe intuitively to be the case for a Newtonian model.Where there is more than one future movement pattern (locus) of an object that is compatible with the 2nd and 3rd laws and the conditions in place at time t, and one or more of those loci involves the object's velocity remaining constant for the period [t,t+h) for some h>0, the pattern that occurs will be one of those latter loci. — andrewk
(Or, equivalently, in a time-reversed scenario, it would take an infinite amount of time for a ball sent sliding up to come to rest at the apex). — Pierre-Normand
It may be that the solution lies in what's being neglected by the problem itself
— creativesoul
Two different questions are being confused here.
The OP was not intended to be about Norton's dome and its claims of Newtonian indeterminism due to a latent jounce concealed in the initial conditions. The OP was about how we would think about an initiating cause when it comes to spontaneous symmetry breaking. — apokrisis
Another way to avoid bifurcations, both future and past, might be to replace the 1st law by a new law saying that the relationship of any force to time must be analytic. — andrewk
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