Newton and the Billiard Ball (DRAFT)

Mark Wilson

University of Pittsburgh

Buttercup: Things are seldom what they seem,

Skim milk masquerades as cream;

Highlows pass as patent leathers;

Jackdaws strut in peacock's feathers.

Captain: Incomprehensible as her utterances are, I nevertheless feel that they are dictated by a sincere regard for me.

W.S. Gilbert, H.M.S. Pinafore

(i)

In the development of science, “simple things” are seldom what they seem. Often fruitful science takes its historical origins within attempts to deal with the swampiest and most intractable forms of descriptive problem. One of the strangest aspects of the Newtonian philosophical heritage stems from the fact that the Principia is often credited with descriptive achievements that it did not, in fact, achieve. In particular, popular writings commonly characterize “Newtonian physics” as “billiard ball mechanics” or as supplying the physical principles relevant to a “clockwork universe.” Neither characterization is accurate and hasty presumptions otherwise encourage a number of unfortunate misapprehensions within philosophical circles that persist with us even today. In this note I shall concentrate upon the pool table side of the ledger.

Let us begin with Robert Boyle’s well-known extolment of “the excellency of the mechanical hypothesis:

The ...thing which recommends the corpuscular principles is their extensiveness. The genuine and necessary effect of the strong motion of one part of matter against another is either to drive it on, in its entire bulk, or to break and divide it into particles of a determinate motion, figure, size, posture, rest, order or texture.[1]

However, familiarity does not breed comprehension. Every modern material scientist knows that the physical issues underlying cohesion and fracture are extremely tricky and generally require descriptive mathematics of considerable complexity. In these regards, a compatriot such as John Locke voiced a more prudent skepticism:

The little Bodies that compose that Fluid, we call Water, are so extremely small, that I have never heard of any one, who by a Microscope, (and yet I have heard of some, that have magnified to 10000; nay, to much above 100,000 times) pretended to perceive their distinct Bulk, Figure, or Motion: And the Particles of Water are also so perfectly loose one from another, that the least force sensibly separates them. Nay, if we consider their perpetual motion, we must allow them to have no cohesion one with another; and yet but a sharp cold come, and they unite, they consolidate, these little Atoms cohere, and are not, without great force, separable. He that could find the Bonds, that tie these heaps of loose little Bodies together so firmly; he that could make known the Cement that makes them stick so fast one to another, would discover a great, and yet unknown Secret: And yet when that was done, would he be far enough from making the extension of Body (which is the cohesion of its solid parts) intelligible, till he could shew wherein consisted the union, or consolidation of the parts of those Bonds, or of that Cement, or of the least Particle of Matter that exists. Whereby it appears that this primary and supposed obvious Quality of Body, will be found, when examined, to be as incomprehensible, as anything belonging to our Minds, and a solid extended Substance, as hard to be conceived, as a thinking immaterial one, whatever difficulties some would raise against it.[2]

However, billiard balls only rarely shatter; indeed, they show a remarkable ability to regain their accustomed shapes swiftly after a considerable degree of buffeting. To a considerable degree of descriptive success, their two body interactions satisfy the simple rules of rebound laid down by Wren and Huygens,[3]which, in a Physics 101 text of today, rely upon two basic conservation principles: the conservation of linear momentum and the conservation of energy (under the assumption that its manifestation is entirely kinetic). Appealing to these principles alone, two balls of equal mass can be predicted to exchange velocities upon collision, maintaining the vectorial sum of their respective individual momentums as they do so. Such expectations are sometimes dubbed the “laws of elastic impact” in philosophical commentary and Descartes is often patronized for getting these “laws” wrong in his Principles of Philosophy.

This popular evaluation strikes me as very misleading, simply because the notion that that anything deserves the title of a “law of impact” is problematic. In reality, colliding balls internally distort in a very complicated way over a short temporal interval, which is ignored within the Wren-Huygens account. Newton himself was aware of many of these behavioral complexities and did not wholeheartedly subscribe to the simple picture just articulated:

Only those bodies which are absolutely hard are exactly reflected according to these rules. Now the bodies here amongst us (being an aggregate of smaller bodies) have a relenting softness and springiness, which makes their contact be for some time and in more points than one. And the touching surfaces during the time of contact do slide one upon another more or less or not at all according to their roughness. And few or none of these bodies have a springiness so strong as to force them one from another with the same vigor that they came together.[4]

(Later in the essay we’ll look at how Newton approached the bodies that “are not here among us,” for these are the “hard atoms” that he posits in the Opticks and elsewhere).

In truth, the standard treatment of billiard ball impact engages in a prudent form of descriptive evasion (= cheating), for incoming and outgoing events are connected together in a black box manner, ignoring the evanescent complications occurring within the actual interval of contact. In modern jargon, treatments of this character are said to sew together “far field” events using asymptotically matching, often employing simple corrective factors as stand-ins for the complications of the true physical interactions. Newton’s own innovation in this vein was to introduce a “coefficient of restitution” factor that attempts to codify the percentage of incoming kinetic energy will be maintained in the outgoing motion. Such descriptive ploys are remarkably effective in the case of billiard balls. Combined with a similar empirical factor due to Euler (that estimates how much the original energy will convert to angular spin), most run-of-the-mill pool table events can be adequately handled (a few exceptions will be discussed below).[5]

Asymptotic workarounds of this type are common in science, even today. Here’s an example that Descartes confronted. He believed that matter was essentially granular, with no gaps appearing between adjacent particles. If so, how can a fluid composed of such particles navigate through an abrupt constriction within a pipe, as illustrated? His rather surprising answer maintains that in the region of the constriction, water particles must rapidly fracture and rejoin in a manner so complicated that the finite human mind cannot follow their details. For Descartes, such “regions of indefiniteness” are fairly common in Nature. Despite these obstacles we can still connect together the upstream and downstream branches of the flow through the assumption that two vital qualities will be conserved throughout, viz., (i) total mass and (ii) Cartesian “quantity of motion,” which is roughly akin to a scalar version of our vectorial linear momentum (∑mv). This assumption allows Descartes to auger the altered velocity within the downstream flow.

However, typical Physics 101 primers that treat billiard ball collisions in a “two conservation laws” manner rarely acknowledge that they are engaging in asymptotics to a complex scattering problem. Michael Spivak, in his refreshing Physics for Mathematicians, captures the conjuring trick nicely:

Elementary physics textbooks need to provide problems that have answers, of course, so, in the manner of a host nonchalantly introducing a celebrity at a party, they will often unobtrusively insert a new definition: a collision is called “completely elastic”, if we also have conservation of kinetic energy… Consorting with this new definition we have a contrasting one: a collision is “completely inelastic” if … the two bodies stick together.…. But having a definition of the coefficient of restitution hardly tells us anything... We would like to understand why the modern definition of a completely elastic collision amounts to an idealization of the concept that lurks in the back of our minds when we think of an “elastic” body as one that pops back into shape after being squashed in a collision.[6]

In fact, a rational explanation of the rebound needs to be far more complicated and must consider the pressure waves that are created by the initial impact, an event that broadens the contact region from a singular point to a wider common interface (mathematically this becomes a so-called “moving boundary value problem”). These initial events send stress waves through the interiors of each ball until they are reflected off the back walls and refocused upon the impact region adequately enough to push the two balls away from one another. The large degree of pressure wave refocusing witnessed in a conventional billiard ball relies heavily upon its spherical shape, the fact that the waves initially spread from a very compact region and the fact that billiard ball “ivory” can swiftly transmit significant pressure waves without significant conversion to heat or suffering permanent plastic distortion. Nonetheless, a small measure of frictional resiliency is required at the interfaces to swiftly pull each ball back to their natural undistorted shapes without permanent wobbling. This resumption of natural rest state will prove a significant factor in the discussion to follow.

The mathematical notions required to formulate these interior wave expectations coherently were not available during the seventeenth century and there was no hope of simulating their complexities in numerical terms until fast enough computers became available recently. As we shall see, these obstacles didn’t prevent a Leibniz from recognizing the basic contours of the mathematical task required.

In any event, a closer inspection of slightly more complicated circumstances shows that the apparent simplicity of the canonical “two conservation law” explanation of billiard ball rebound represents a derivational illusion fueled by what we might call special assumption channeling. In particular, consider the familiar pendulum toy often called Newton’s cradle (illustrated). At first blush, the gizmo’s behaviors appear to neatly underwrite the usual “two conservation law” predictions. Thus if we begin with two balls pulled off to the left, then two balls will depart on the right, leaving the central balls motionless, just as our two equations would have predicted. But this prediction works only if there is a small gap between the balls that allows each pairwise collision to run through its compressive cycles with extreme swiftness before the struck ball collides with its neighbor to the left. When these conditions are not met, the ensemble will often wiggle and divide in unexpected manners that can only be explained by computing the locations where the pressure waves passing through several conjoined balls will refocus in a manner that can drive the balls apart.[7]

What do I mean by “special assumption channeling”? In a generic case of impactive interaction—two blocks of wood banging together, say--, a large number of descriptive variables are required, demanding a comparable number of governing equations to account for their interactions. By assigning our billiard balls wholly rigid shapes sliding upon a rigid plane, tolerating no internal potential energy storage and reducing their interface to a single point of normal contact, our two conservation laws allow us to predict the outgoing motions through Wren-Huygens asymptotics as above. Although the same two equations remain valid for our pieces of wood, we will require many further equations to resolve their behaviors adequately, due to the fact that we can’t reduce the pertinent descriptive variables so drastically as we can in the special circumstances of two billiard balls that clash head on. Allied observations explain why a Newton’s cradle will “act properly” only if small gaps secretly separate the component balls.

Later we will observe that the “explanatory illusions” engendered by tacit channeling assumptions of this character have played a significant role in generating significant forms of philosophical confusion.

(ii)

In light of this complexity, why did the scientists of the early modern era focus so intently much upon impactive events like billiard collisions?

Like Boyle, they presumed that most physical processes operate through contact action and hence “simple” collisions between bodies that contact one another locally (e.g., at the solitary junction point between two rigid spheroids) should be investigated as emblematic of nature operating in its most elementary and foundational manner, in terms of which its more complicated entanglements could be then decomposed. The methodological percept adopted is “Find the fundamental solutions first and build up from there.”

In this spirit, these scientists naturally presumed that when billiard ball A directly affects ball B through head-on contact, some motive capacity must be transferred across the bodies that should supply the proper measure of the amount of “force” transferred. Thus the notion of a singularly acting impulse becomes central, where an “impulse” represents a primitive activity that transpires both locally and instantaneously. Even Newton adopted these priorities, for he conceptually decomposed a smoothly acting force such as gravity into a rapid sequence of impactive hammer blows.

Mathematically, however, such “fundamental solution” reductions have often proved problematic, at least from the point of view of tractable mathematical description. This is because impactive events inherently represent singular episodes where important rates of change become ill-defined; smoothly and continuously acting forces suit the tools of the calculus far better. A familiar historical illustration of this dilemma can be found in the Euler-d’Alembert dispute over the “fundamental solution” for a loaded string. Euler maintained that more complicated loading arrangements could be compounded from the superposition of simple triangular string distortions as illustrated. But d’Alembert reasonably objected, how can such a “solution” coherently satisfy the pertinent governing equation for its internal physics requires that the string balance the gravitation pull by curving? In the calculus, “curving” is captured by the second derivative ∂2y/∂x2 but no bending of that sort can be found anywhere within Euler’s so-called “fundamental solution.”