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The Norton dome is a dome-shaped surface in a gravity field on the top of which a symmetrical object is placed in perfect balance.
According to Newtonian determinism, the object will remain in balance until an external agency supplies force and the object can slide down.
On a Norton dome though, the balanced state of the object can be disturbed. There is a probability it will slide down the dome. Not because of an applied force but because the state of balance is undetermined.

It's the special shape of the dome that induces this undetermined state. Somehow the particle on top is prone to slide down without a disturbance of the forces acting on it. A particle in balance on a sphere or on top of a vertical line will remain there forever. The shape is important.

What is happening when the particle slides down? How can it slide down when all forces are equally pulling on it? When the particle is displaced an infinitesimal distance from balance, all forces cease to pull on it. Only the force in the direction of the displacement remains. But that holds too for the sphere or a pyramid. So what makes the dome special? What's got the Norton dome other domes haven't?

MatterGauge
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1 Answers1

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How can it slide down when all forces are equally pulling on it?

Would it slide down if it had an initial velocity in a certain direction? Of course, it would! Now, of course, in Norton's dome, it is not the case that it has an initial velocity in either direction. But, the point to recognize here is as to what constitutes a stringent enough initial condition that the equations of motion guarantee a unique time-evolution for the given initial condition. Under usual potentials, the position and the velocity of a particle constitute such initial conditions that the equations of motion ensure a unique time-evolution of the particle. However, for singular potentials, such as the one encountered in Norton's dome, this is no longer true. So, there simply isn't a unique prediction that we can make about the time-evolution of a particle sitting on top of Norton's dome. Now, that does not mean that the theory predicts intrinsic probabilities for what would happen, it simply says that the initial conditions of the position and the velocity are not good enough to predict the time-evolution of the particle.

So what makes [Norton's] dome special?

What makes it special is that the potential for Norton's dome is singular at the top, in particular, it goes as $\propto s^{3/2}$ where $s$ is the radial distance from the top. It is singular in the sense that $V''(s=0)$ is divergent. This is what makes it so that the initial velocity and the initial position do not constitute a stringent enough initial condition for the equations of motion to predict a unique time-evolution. There is another way to look at it. If you were to throw a particle towards the top of a dome in such a way that it reaches the top with precisely zero velocity then its time-reversal would say that the time-evolution of a particle sitting on top of the dome is not unique. In usual domes, it would take an infinite amount of time for a particle to reach the top of the dome if you throw it with just enough velocity to reach the top with zero velocity. However, in the case of Norton's dome, the shape is such that it would reach the top of the dome in a finite time -- making the time-reversed version of the motion indeterminate.