Abstract
If the force on a particle fails to satisfy a Lipschitz condition at a point, it relaxes one of the conditions necessary for a locally unique solution to the particle’s equation of motion. I examine the most discussed example of this failure of determinism in classical mechanics—that of Norton’s dome—and the range of current objections against it. Finding there are many different conceptions of classical mechanics appropriate and useful for different purposes, I argue that no single conception is preferred. Instead of arguing for or against determinism, I stress the wide variety of pragmatic considerations that, in a specific context, may lead one usefully and legitimately to adopt one conception over another in which determinism may or may not hold.
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Notes
He did not, however, investigate how ubiquitous such systems might be.
See Hoering (1969) for an overview of the terms of that debate, which focused mostly on issues of predictability.
Norton originally concocted the example as part of an argument against causation as an underlying principle in modern physics, including classical mechanics, but this claim will not play a role in the rest of this paper.
I will use “classical” and “Newtonian” as synonyms unless otherwise specified.
One typically proves this first for first-order ordinary differential equations, then extends it to higher-order equations. See, for example, Arnol’d (1992, p. 36–38, 104–105).
The locality of this kind of determinism can be highlighted by reference to the “space invader” scenarios alluded to in Section 1, in which the specification of the initial conditions of the system in question fixes its history for only a finite period of time (Earman 2007, 2008). But this locality as such will not play a role in the examples of this paper.
To see why, suppose otherwise. Then there must be some constant K > 0 satisfying |F(r)| ≤ K|r|. But any choice of r satisfying 0 < r < (mb 2/K)1 − a yields a contradiction.
A function f : ℝ → ℝn is smooth when its derivatives \(\frac{d^n f}{dt^n}\) exist and are continuous for each n ∈ ℕ.
The “time-reversal” here is that of the equations of motion, rather than of the laws of Newtonian mechanics. But really one does not even need that since it is easy to reverse-engineer the appropriate choice of initial velocity to bring the particle to rest at the apex given an initial location on the surface of the dome.
In the quoted passage Poisson is speaking directly about solutions to the differential equation \(dv/dt = -a \sqrt{v}\), but in his later remarks on d 2 x / dt 2 = ax n, of which Norton’s dome is a special case, he stresses the same conclusions more briefly.
Cf. Vickers (2008) on a recent debate about the consistency of classical electrodynamics, in which he argues the parties involved have largely been talking past each other. Unlike Vickers, however, for the purposes of this paper I remain agnostic about whether Norton’s formulation of classical mechanics is interesting or important. Recently Vickers (2011) has synthesized the morals from both debates to urge discussants not to talk past each other on account of having different conceptions of some theory, a proposal to which I am sympathetic.
Cf. Vickers (2009) on the consistency of Newtonian cosmology prior to mathematical and conceptual developments of the late nineteenth and early twentieth centuries.
One might consider a force function f not Lipschitz continuous on some open subset A of its domain. But then f is not continuously differentiable simpliciter on A (see Arnol’d 1992, p. 273). One can extend this result to show that, in fact, such an f cannot even be differentiable on A, for the points of continuity of any derivative must be dense and G δ in A, excluding the empty set (see theorem 2.2 of Bruckner 1978).
Norton (2008, p. 790) found this particular example independently and mentions it briefly. Zimba (2008) developed a somewhat similar electrostatic example using an ideal quadrupole and a spherical charge distribution, except with a pair of conical sections symmetric about the z-axis removed. He made this construction to avoid any objections regarding “collisions” between the charge distribution and the point particle—an objection, as I outline below, I think is weak—but as a result the force on the point particle is proportional to \(\sqrt{|z|}\) only to first order. Thus Zimba’s example is not clearly one of Lipschitz indeterminism.
The particle then undergoes simple harmonic motion. See, for example, Resnick (1992).
It is plausible to account for the effect of any radiation emitted by the point particle on its motion similarly: allowing for such radiation does not undermine the example as long as the particle begins moving at all. It is even better, however, to remind oneself that taking the point particle to be a test charge means that one does not consider it as an electromagnetic source. Hence it does not interact with its own electromagnetic field or experience a self-force, just as the point particle in the case of Norton’s dome does not interact with its own gravitational field or experience a self-force.
An uncannily careful reader might observe that, to be precise, the derivation of this distribution involves taking a disk whose thickness is negligible compared with its radius; then half of Eq. 16 gives the charge distribution on each side.
An ideal dipole is a formed as follows. Take two point charges q and − q, and let d be the displacement vector from − q to q. Define the dipole moment as p = q d, then take the limit as q → ∞ and ||d|| → 0 while holding p constant. Note that p ≠ 0, for then otherwise q = 0.
There is an additional issue with the ideal dipole, as any dipole with non-zero physical size will not suffice for Lipschitz indeterminism.
He acknowledges that “they split into further subdivisions as further questions are pressed” (Wilson 2009, p. 176) so that, depending on how fine-grained one would like to be, there may be many more kinds of classical mechanics than just three.
It is clear that such descriptive holes are not sufficient for Lipschitz indeterminism, and the electrostatic examples, which use only point particles, show it is not necessary. (See, as well, Roberts (2009)). In addition, Korolev (2008) has constructed Lipschitz indeterministic examples in continuum mechanics.
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Acknowledgements
Thanks to Jeff Barrett, David Malament, Peter Vickers, Jim Weatherall, and two anonymous referees for helpful comments, and to the audiences of both the Southern California Philosophy of Physics Group and the Sixth Logic, Mathematics, and Physics Graduate Philosophy Conference at the University of Western Ontario for comments on earlier versions. Thanks also to Jennifer C. Herrera for comments on my translation of Poisson. Part of the present work was written with the support of a National Science Foundation Graduate Research Fellowship.
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Fletcher, S.C. What counts as a Newtonian system? The view from Norton’s dome. Euro Jnl Phil Sci 2, 275–297 (2012). https://doi.org/10.1007/s13194-011-0040-8
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DOI: https://doi.org/10.1007/s13194-011-0040-8