Norton’s Dome (Column 687)

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A brief look at determinism

I have often discussed here the question of determinism: a view according to which the present state fixes the future uniquely—i.e., it cannot be that the same history yields two different futures. This touches on the human being and our relation to the rest of nature: it is commonly thought that “nature” (inanimate, vegetative, animal) is determinist; the open question is only the human being—does a person have free will?

From the early 20th century, quantum theory developed and exhibited non-deterministic domains (at least according to the common interpretations). Chaos theory came later and is often (mistakenly) presented as undermining determinism; but I have written more than once that chaos does not deny determinism. What it presents is a computational difficulty in predicting the future; the future itself, given complete data, is uniquely determined. By contrast, in quantum theory—again, at least in the common interpretations—it appears there really is a departure from determinism.

As for classical physics—once one sets chaos aside—there is wall-to-wall agreement that it is entirely determinist.

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Determinism and reversibility in Newtonian mechanics

The laws of physics—and in particular Newtonian mechanics—are described by differential equations that determine the next state from the present one. In effect, given complete knowledge of the present (and sufficient computational power), one can fix the future categorically.

For those unfamiliar with the derivative, here is a brief reminder. Suppose X(t)X(t) records, say, the position of a body as a function of time. We may ask about the rate of change of XX with time: how fast it changes, and in which direction. The derivative quantifies this:

(1)

If during seconds the position changes by meters, the average rate is m/s. Passing to ever smaller gives the instantaneous rate. Differentiating once more gives the acceleration ()—the rate of change of velocity.

Newton’s second law ties this to force:

(2)

Here is mass, is position, and is the force (which may depend on position and on time). Solving such an equation means finding the function —the full trajectory. To do so uniquely one must add initial conditions: the position and velocity at some initial time. With those specified, the mechanics is (classically) fully deterministic.

A further feature is time reversibility: if we film a Newtonian process and run the film backwards, we get another legitimate Newtonian process. The equations are second-order in time; replacing by leaves the structure intact.

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Norton’s Dome

Now to the “dome.” Consider a perfectly symmetric dome whose radial cross-section is given (in cylindrical coordinates) by

(3)

Here is the horizontal distance from the apex along the surface, is the vertical height (measured downward from the apex), is the gravitational acceleration, and is a positive constant.

Place a frictionless point-mass (“a tiny bead”) at rest exactly at the apex. By symmetry, the tangential component of gravity along the surface turns out to be proportional to , so the radial equation of motion (again, idealizing: no friction, only gravity) is:

(4)

Impose the natural initial conditions of “rest on the top”:

It turns out there are two kinds of solutions:

(5a) (the bead stays forever at the apex);

(5b)

for any arbitrary .

Interpretation: the bead can wait atop the dome for an arbitrary duration and then, with no external trigger and from zero velocity, spontaneously begin to slide down. Because of the dome’s rotational symmetry it may do so in any horizontal direction. Thus there are infinitely many solutions: for every choice of and every direction.

By time-reversal, one also gets the mirror family: a bead that comes up the dome from some direction, slows down exactly at the apex, sits there for an arbitrary time, and (run backwards) “came from” a spontaneous start at some earlier .

Why doesn’t this happen on a spherical dome? On a true sphere, to arrive at rest exactly at the top requires an ascent that takes infinite time; conversely, to start from exact rest on top would require rewinding from an infinite past—there is no finite-time “takeoff.” Norton’s special profile (3) avoids that and permits finite-time departure.

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Why this is a paradox

This looks like a direct contradiction of Newtonian determinism. We supplied perfectly good initial data (), yet multiple futures are possible: the bead may remain forever, or depart at any time in any direction. Nothing in the system “chooses” or the direction; it is as if there is spontaneous motion “for no reason.”

How to respond to a paradox? In general, we have a few options:

1. Find a mistake in the reasoning.

2. Reject one of the assumptions.

3. Give up the contested conclusion.

4. Or admit tzarich iyyun (“needs investigation”) and suspend judgment.

Here, the mathematics looks correct; and even if one were to reject Newton’s laws as a true description of nature, the paradox arises from the equations themselves, not from experiment. So the third path—abandoning determinism derived from Newton—is the one the dome seems to force: perhaps Newton’s laws (despite being differential equations) are not always determinist.

For discussion of background and related issues (Hebrew), see the Q&A here:

https://mikyab.net/%D7%A9%D7%95%D7%AA/%D7%93%D7%98%D7%A8%D7%9E%D7%99%D7%A0%D7%99%D7%96%D7%9D-%D7%95%D7%94%D7%9B%D7%99%D7%A4%D7%94-%D7%A9%D7%9C-%D7%A0%D7%95%D7%A8%D7%98%D7%95%D7%9F/

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Is determinism really just a corollary of Newton?

What makes Norton’s example so unsettling is that determinism is not merely a reading of Newton’s equations—it is also tied to causality: nothing happens without a cause. If Newton’s laws sometimes fail to be determinist, then even if they are a decent approximation, they cannot be the correct description of the physical world, which (at least classically) we expect to be causal and determinist. Events should not occur without causes; the time and direction of the bead’s motion should be the result of some cause, not arbitrary.

Yes, quantum theory suggests there may be genuinely non-causal phenomena (depending on interpretation—hidden variables, non-local causality, etc.). But our classical intuition about causality remains powerful. If we don’t give it up quickly for quantum mechanics, all the more so we should not abandon it for classical mechanics.

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Common “resolutions”

One often-noted point (see the literature) is that at the force law embodied in (4) is not Lipschitz/“nice” enough; uniqueness theorems for ODEs fail, and the non-uniqueness of solutions is mathematically allowed. But that observation alone does not solve the physical problem; at most it says: Newton’s equations, taken with such potentials/shapes, need not be determinist—and thus cannot be a faithful representation of nature in such cases.

A different line (raised in the discussion I saw) is to rewrite Newton’s second law as: “When no force acts, there is no acceleration; when a force acts, there is.” Up to the bead is stationary at the apex and the tangential component vanishes, so there; past , the force becomes nonzero and motion begins. But this does not explain how the system passes from the apex (where the tangential component is zero) to any neighboring point where it is nonzero, without something that causes that departure. The step where the law “turns on” remains opaque.

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“There is no square since the Six Days of Creation”: a quick fix?

My initial thought was that the issue lies in the shape: the function defining the dome has a cusp at the apex so that second derivatives (curvatures) misbehave there. Perhaps such a shape cannot exist in the physical world: nature, so to speak, does not produce true cusps and second-derivative discontinuities.

Said differently: to speak of an object with perfect, cusp-like features requires arbitrarily fine spatial resolution; but at sufficiently fine scales, the classical continuum picture breaks down and quantum/atomic granularity rules. In that regime, classical mechanics does not apply; the analysis is simply not about the physical world. In a “rounded” reality without perfect cusps, the paradoxical behavior would not arise.

An old rabbinic quip captures the intuition: “Since the Six Days of Creation, there is no (perfect) square”—i.e., nature yields rounded, differentiable forms; sharp corners are the work of intentional craftsmanship, not spontaneous nature (cf. Tosefta Ma’aserot ch. 3; Yerushalmi Ma’aserot 5:3). Even where nature seems to make peaks, on closer inspection the second derivative exists and behaves—our choice of variables may create apparent non-smoothness, but nature itself “likes” continuity and differentiability. See also the mathematical notion of density in number theory for an analogy of “no nearest point” on a continuum: https://he.wikipedia.org/wiki/%D7%A6%D7%A4%D7%99%D7%A4%D7%95%D7%AA_(%D7%AA%D7%95%D7%A8%D7%AA_%D7%94%D7%9E%D7%A1%D7%A4%D7%A8%D7%99%D7%9D)

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Bottom line

It seems we can retain: (a) the causal–determinist intuition for classical physics; and (b) Newton’s laws as an excellent description of classical reality—provided we exclude idealized shapes/conditions that cannot occur in the physical world (true cusps, perfect non-Lipschitz features, etc.). That is already a substantial conclusion, but a digestible one.

For a related earlier column, see here:

https://mikyab.net/translated-articles-rabbi-michael-abraham/post-83362