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The dome equation is expressed in terms of arc length and height, which hides a lot of bad behaviour. As Luboš has pointed out, beyond a certain point it is no longer physical so we need to constraint it before that happens.

You can actually work out the Cartesian equation for it, which is plottable:

This isn't the only constraint that should be applied though. It's said that because of the non-analytical nature of the dome curve (it's not differentiable at the apex beyond the first derivative) it admits more than one solution. In fact, it's the square root that allows us to easily intuit the other solution which represents the trajectory of a ball reaching or leaving the apex with velocity $v$ at the limit $v = 0$.

Norton tries to pass off his other solution as Newtonian but in fact it's not (at the apex, anyway). This in itself isn't the source of his claim of non-determinism though. That comes from him stitching two solutions together that have different initial conditions at some arbitrary time T. This makes zero sense and has no physical justification.

I've written a detailed analysis of why Norton's dome does not prove Newtonian mechanics is non-deterministic here.

The dome equation is expressed in terms of arc length and height, which hides a lot of bad behaviour. As Luboš has pointed out, beyond a certain point it is no longer physical so we need to constraint it before that happens.

You can actually work out the Cartesian equation for it, which is plottable:

This isn't the only constraint that should be applied though. It's said that because of the non-analytical nature of the dome curve (it's not differentiable at the apex beyond the first derivative) it admits more than one solution. In fact, it's the square root that allows us to easily intuit the other solution which represents the trajectory of a particle reaching or leaving the apex with velocity $v$ at the limit $v = 0$.

Norton tries to pass off his other solution as Newtonian but in fact it's not (at the apex, anyway). This in itself isn't the source of his claim of non-determinism though. That comes from him stitching two solutions together that have different initial conditions at some arbitrary time T. This makes zero sense and has no physical justification.

I've written a detailed analysis of why Norton's dome does not prove Newtonian mechanics is non-deterministic here.

The dome equation is expressed in terms of arc length and height, which hides a lot of bad behaviour. As Luboš has pointed out, beyond a certain point it is no longer physical so we need to constraint it before that happens.

You can actually work out the Cartesian equation for it, which is plottable:

This isn't the only constraint that should be applied though. It's said that because of the non-analytical nature of the dome curve (it's not differentiable at the apex beyond the first derivative) it admits more than one solution. In fact, it's the square root that allows us to easily intuit other solutions. Any even power of t higher than 2 is a solution too. None of them are Newtonian though and all violate energy conservation laws.

Norton tries to pass his other solution as Newtonian but in fact it's not. This in itself isn't the source of his claim of non-determinism though. That comes from him stitching two solutions together that have different initial conditions at some arbitrary time T. This makes zero sense and has no physical justification.

I've written a detailed analysis of why Norton's dome does not prove Newtonian mechanics is non-deterministic here.

The dome equation is expressed in terms of arc length and height, which hides a lot of bad behaviour. As Luboš has pointed out, beyond a certain point it is no longer physical so we need to constraint it before that happens.

You can actually work out the Cartesian equation for it, which is plottable:

This isn't the only constraint that should be applied though. It's said that because of the non-analytical nature of the dome curve (it's not differentiable at the apex beyond the first derivative) it admits more than one solution. In fact, it's the square root that allows us to easily intuit the other solution which represents the trajectory of a ball reaching or leaving the apex with velocity $v$ at the limit $v = 0$.

Norton tries to pass off his other solution as Newtonian but in fact it's not (at the apex, anyway). This in itself isn't the source of his claim of non-determinism though. That comes from him stitching two solutions together that have different initial conditions at some arbitrary time T. This makes zero sense and has no physical justification.

I've written a detailed analysis of why Norton's dome does not prove Newtonian mechanics is non-deterministic here.

The dome equation is expressed in terms of arc length and height, which hides a lot of bad behaviour. As Luboš has pointed out, beyond a certain point it is no longer physical so we need to constraint it before that happens.

You can actually work out the Cartesian equation for it, which is plottable:

This isn't the only constraint that should be applied though. Because of the non-analytical nature of the dome curve (it's not differentiable at the apex beyond the first derivative) it admits more one solution.

Norton tries to pass his other solution as Newtonian but in fact it's not. This in itself isn't the source of his claim of non-determinism though. That comes from him stitching two solutions together that have different initial conditions at some arbitrary time T. This makes zero sense and has no physical justification.

I've written a detailed analysis of why Norton's dome does not prove Newtonian mechanics is non-deterministic here.

The dome equation is expressed in terms of arc length and height, which hides a lot of bad behaviour. As Luboš has pointed out, beyond a certain point it is no longer physical so we need to constraint it before that happens.

You can actually work out the Cartesian equation for it, which is plottable:

This isn't the only constraint that should be applied though. It's said that because of the non-analytical nature of the dome curve (it's not differentiable at the apex beyond the first derivative) it admits more than one solution. In fact, it's the square root that allows us to easily intuit other solutions. Any even power of t higher than 2 is a solution too. None of them are Newtonian though and all violate energy conservation laws.

Norton tries to pass his other solution as Newtonian but in fact it's not. This in itself isn't the source of his claim of non-determinism though. That comes from him stitching two solutions together that have different initial conditions at some arbitrary time T. This makes zero sense and has no physical justification.

I've written a detailed analysis of why Norton's dome does not prove Newtonian mechanics is non-deterministic here.