3.2 Natural and Violent Motions 

Mr Spenser in the course of his remarks regretted that so many members of the Section were in the habit of employing the word Force in a sense too limited and definite to be of any use in a complete theory. He had himself always been careful to preserve that largeness of meaning which was too often lost sight of in elementary works. This was best done by using the word sometimes in one sense and sometimes in another, and in this way he trusted that he had made the word occupy a sufficiently large field of thought. 
James Clerk Maxwell 

The concept of force is one of the most peculiar in all of physics. It is, in one sense, the most viscerally immediate concept in classical mechanics, and seems to serve as the essential "agent of causality" in all interactions, and yet the ontological status of force has always been highly suspect. We sometimes regard force as the cause of changes in motion, and imagine that those changes would not occur in the absence of the forces, but this causative aspect of force is an independent assumption that does not follow from any quantifiable definition, since we could equally well regard force as being caused by changes in motion, or even as merely a descriptive parameter with no independent ontological standing at all. 

In addition, there is an inherent ambiguity in the idea of changes in motion, because it isn't obvious what constitutes unchanging (i.e., unforced) motion. Aristotle believed it was necessary to distinguish between two fundamentally distinct kinds of motion, which he called natural motions and violent motions. The natural motions included the apparent movements of celestial objects, the falling of leaves to the ground, the upward movement of flames and hot gases in the atmosphere, or of air bubbles in water, and so on. According to Aristotle, the cause of such motions is that all objects and substances have a natural place or level (such as air above, water below), and they proceed in the most direct way, along straight vertical paths, to their natural places. The motion of the celestial bodies is circular because this is the most perfect kind of unchanging eternal motion, whereas the necessarily transitory motions of sublunary objects are rectilinear. It may not be too misleading to characterize Aristotle's concept of sublunary motion as a theory of buoyancy, since the natural place of light elements is above, and the natural place of heavy elements is below. If an object is out of place, it naturally moves up or down as appropriate to reach its proper place. 

Aristotle has often been criticized for saying (or seeming to say) that the speed at which an object falls (through the air) is proportional to its weight. To the modern reader this seems absurd, as it is contradicted by the simplest observations of falling objects. However, it's conceivable that we misinterpret Aristotle's meaning, partly because we're so accustomed to regarding the concept of force as the cause of motion, rather than as an effect or concomitant attribute of motion. If we consider the downward force (which Aristotle would call the weight) of an object to be the force that would be required to keep it at its current height, then the "weight" of an object really is substantially greater the faster it falls. More strength is required to catch a falling object than to hold the same object at rest. Some Aristotelian scholars have speculated that this was Aristotle's actual meaning, although his writing's on the subject are so sketchy that we can't know for certain. In any case, it illustrates that the concept and significance of force in a physical theory is often murky, and it also shows how thoroughly our understanding of physical phenomena is shaped by the distinction between forces (such as gravity) that we consider to be causes of motion, and those (such as impact forces) that we consider to be caused by motion. 

Aristotle also held that the speed of motion was not only proportional to the "weight" (whatever that means) but inversely proportional to the resistance of the medium. Thus his proposed law of motion could be expressed roughly as V = W/R, and he used this to argue against the possibility of empty space, i.e., regions in which R = 0, because the velocity of any object in such a region would be infinite. This doesn't seem like a very compelling argument, since we could easily counter that the putative vacuum would not be the natural place of any object, so it would have no "weight" in that direction either. Nevertheless, perhaps to avoid wrestling with the mysterious fraction 0/0, Aristotle surrounded the four sublunary elements of Earth, Water, Air, and Fire with a fifth element (quintessence), the lightest of all, called aether. This aether filled the superlunary region, ensuring that we would never need to divide by zero. 

In addition to natural motions, Aristotle also considered violent motions, which were any motions resulting from acts of volition of living beings. Although his writings are somewhat obscure and inconsistent in this area, it seems that he believed such beings were capable of selfmotion, i.e., of initiating motion in the first instance, without having been compelled to motion by some external agent. Such selfmovers are capable of inducing composite motions in other objects, such as when we skip a stone on the surface of a pond. The stone's motion is compounded of a violent component imparted by our hand, and the natural component of motion compelling it toward its natural place (below the air and water). However, as always, we must be careful not to assume that this motion is to be interpreted as the causative result of the composition of two different kinds of forces. It was, for Aristotle, simply the kinematic composition of two different kinds of motion. 

The bifurcation of motion into two fundamentally different types, one for natural motions of nonliving objects and another for acts of human volition – and the attention that Aristotle gave to the question of unmoved movers, etc. – is obviously related to the issue of free will, and demonstrates the strong tendency of scientists in all ages to exempt human behavior from the natural laws of physics, and to regard motions resulting from human actions as original, in the sense that they need not be attributed to other motions. We'll see in Section 9 that Aristotle's distinction between natural and violent motions plays a key role in the analysis of certain puzzling aspects of quantum theory. 

We can also see that the ontological status of "force" in Aristotle's physics is ambiguous. In some circumstances it seems to be more an attribute of motion rather than a cause of motion. Even if we consider the quantitative physics of Galileo, Newton, and beyond, it remains true that "force", while playing a central role in the formulation, serves mainly as an intermediate quantity in the calculations. In fact, the concept of 'force' could almost be eliminated entirely from classical mechanics. (See Section 4.4 for further discussion of this.) Newton wrestled with the question of whether force should be regarded as an observable or simply a relation between observables. Interestingly, Ernst Mach regarded the third law as Newton's most important contribution to mechanics, even though other's have criticized it as being more a definition than a law. 

Newton’s struggle to find the "right" axiomatization of mechanics can be seen by reading the preliminary works he wrote leading up to The Principia, such as "De motu corporum in gyrum" (On the motion of bodies in an orbit). At one point he conceived of a system with five Laws of Motion, but what finally appeared in Principia were eight Definitions followed by three Laws. He defined the "quantity of matter" as the measure arising conjointly from the density and the volume. In his critical review of Newtonian mechanics, Mach remarked that this definition is patently circular, noting that "density" is nothing but the quantity of matter per volume. However, all definitions ultimately rely on undefined (irreducible) terms, so perhaps Newton was entitled to take density and volume as two such elements of his axiomatization. Furthermore, by basing the quantity of matter on explicitly finite density and volume, Newton deftly precluded pointlike objects with finite quantities of matter, which would imply the existence of infinite forces and infinite potential energy according to his proposed inversesquare law of gravity. 

The next basic definition in Principia is of the "quantity of motion", defined as the measure arising conjointly from the velocity and the quantity of matter. Here we see that "velocity" is taken as another irreducible element, like density and volume. Thus, Newton's ontology consists of one irreducible entity, called matter, possessing three primitive attributes, called density, volume, and velocity, and in these terms he defines two secondary attributes, the "quantity of matter" (which we call "mass") as the product of density and volume, and the "quantity of motion" (which we call "momentum") as the product of velocity and mass, meaning it is the product of velocity, density, and volume. Although the term "quantity of motion" suggests a scalar, we know that velocity is a vector, (i.e., it has a magnitude and a direction), so it's clear that momentum as Newton defined it is also is a vector. After going on to define various kinds of forces and the attributes of those forces, Newton then, as we saw in Section 1.3, took the law of inertia and relativity as his First Law of Motion, just as Descartes and Huygens had done. Following this we have the "force law", i.e., Newton's Second Law of Motion: 

The change of motion is proportional to the motive force impressed; and is made in the direction of the right line in which the force is impressed. 

Notice that this statement doesn't agree precisely with either of the two forms in which the Second Law is commonly given today, namely, as F = dp/dt or F = ma. The former is perhaps closer to Newton's actual statement, since he expressed the law in terms of momentum rather than acceleration, but he didn't refer to the rate of change of momentum. No time parameter appears in the statement at all. This is symptomatic of a lack of clarity (as in Aristotle’s writings) over the distinction between "impulse force" and "continuous force". Recall that our speculative interpretation of Aristotle's downward "weight" was based on the idea that he actually had in mind something like the impulse force that would be exerted by the object if it were abruptly brought to a halt. Newton's Second Law, as expressed in the Principia, seems to refer to such an impulse, and this is how Newton used it in the first few Propositions, but he soon began to invoke the Second Law with respect to continuous forces of finite magnitude applied over a finite length of time – more in keeping with a continuous force of gravity, for example. This shows that even in the final version of the axioms and definitions laid down by Newton, he did not completely succeed in clearly delineating the concept of force that he had in mind. Now, in each of his applications of the Second Law, Newton made the necessary dimensional adjustments to appropriately account for the temporal aspect that was missing from the statement of the Law itself, but this was done ad hoc, with no clear explanation. (His ability to reliably incorporate these factors in each context testifies to his solid grasp of the new dynamics, despite the imperfections of his formal articulation of it.) Subsequent physicists clarified the quantitative meaning of Newton’s second law, explicitly recognizing the significance of time, by expressing the law either in the form F = d(mv)/dt or else in what they thought was the equivalent form F = m(dv/dt). In the context of special relativity these two are not equivalent, and only the former leads to a coherent formulation of mechanics. (It’s also worth noting that, in the context of special relativity, the concept of force is largely an anachronism, and it is introduced mainly for the purpose of relating relativistic descriptions to their classical counterparts.) 

The third Law of Motion in the Principia is regarded by many people as one of Newton's greatest and most original contributions to physics. This law states that 

To every action there is always opposed an equal reaction: or, the mutual actions of two bodies upon each other are always equal, and directed to contrary parts. 

Unfortunately the word "action" is not found among the previously defined terms, but in the subsequent text Newton clarifies the intended meaning. He says "If a body impinge upon another, and by its force change the motion of the other, that body also... will undergo an equal change in its own motion towards the contrary part." In other words, the net change in the "quantity of motion" (i.e., the sum of the momentum vectors) is zero, so momentum is conserved. More subtly, Newton observes that "If a horse draws a stone tied to a rope, the horse will be equally drawn back towards the stone". This is true even if neither the horse nor the stone are moving (which implies that they are each subject to other forces as well, tending to hold them in place). The illustrates how the concept of force enables us to conceptually decompose a null net force into nonnull components, each representing the contributions of different physical interactions. 

In retrospect we can see that Newton's three "laws of motion" actually represent the definition of an inertial coordinate system. For example, the first law imposes the requirement that the spatial coordinates of any material object free of external forces are linear functions of the time coordinate, which is to say, free objects move with a uniform speed in a straight line with respect to an inertial coordinate system. Rather than seeing this as a law governing the motions of free objects with respect to a given system of coordinates, it is more correct to regard it as defining a class of coordinate systems in terms of which a recognizable class of motions have particularly simple descriptions. It is then an empirical question as to whether the phenomena of nature possess the attributes necessary for such coordinate systems to exist. 

The significance of “force” was already obscure in Newton’s three laws of mechanics, but it became even more obscure when he proposed the law of universal gravitation, according to which every particle of matter exerts a force of attraction on every other particle of matter, with a strength proportional to its mass and inversely proportional to the square of the distance. The rival Cartesians expected all forces to be the result of local contact between bodies, as when two objects press directly against each other, but Newton’s conception of instantaneous gravity between distant objects seems to defy representation in those terms. In an effort to reconcile universal gravitation with semiCartesian ideas of force, Newton’s young friend Nicolas Fatio hypothesized an omnidirectional flux of small “ultramundane” particles, and argued that the mutual shadowing effect could explain why massive bodies are forced together. The same idea was later taken up by Georges Lesage, but it entails thermodynamic consequences that make it untenable, and moreover an elementary force of attraction would still be required to hold together the parts of the fundamental particles, so the shadow idea does not actually dispense with primitive attractive forces (barring an infinite regress of ultraultramundane particles, etc.). 

The simple notion of force at a distance was so successful that it became the model for all mutual forces (both attractive and repulsive) between objects, and the early theories of electricity and magnetism were expressed in those terms. However, reservations about the intelligibility of instantaneous action at a distance remained. Eventually Faraday and Maxwell introduced the concept of disembodied “lines of force”, which later came to be regarded as fields of force, almost as if force was an entity in its own right, capable of flowing from place to place. In this way the Maxwellians (perhaps inadvertently) restored the Cartesian ideas that all space must be occupied and that all forces must be due to direct local contact. They accomplished this by positing a new class of entity, namely the field. Admittedly our knowledge of the electromagnetic field is only inferred from the behavior of matter, but it was argued that explanations in terms of fields are more intelligible than explanations in terms of instantaneous forces at a distance, mainly because fields were considered necessary for strict conservation of energy and momentum once it was recognized that electromagnetic effects propagate at a finite speed. 

However, the explanation of phenomena in terms of fields, characterized by partial differential equations, was incomplete, because it was not possible to represent stable configurations of matter in these terms. Maxwell’s field equations are linear, so there was no hope of them possessing solutions corresponding to discrete electrical charges or particles of matter. Hence it was still necessary to retain the laws of mechanics of discrete entities, characterized by total differential equations. The conceptual dichotomy between Newton’s physics of particles and Maxwell’s physics of fields is clearly shown by the contrast between total and partial differential equations, and this contrast was seen (by some people at least) as evidence of a fundamental flaw. In a 1936 retrospective essay Einstein wrote 

This is the basis on which H. A. Lorentz obtained his synthesis of Newton’s mechanics and Maxwell’s field theory. The weakness of this theory lies in the fact that it tried to determine the phenomena by a combination of partial differential equations (Maxwell’s field equations for empty space) and total differential equations (equations of motions of points), which procedure was obviously unnatural. 

The difference between total and partial differential equations is actually more profound than it may appear at first glance, because (as alluded to in Section 1.1) it entails different assumptions about the existence of freewill and acts of volition. If we consider a pointlike particle whose spatial position x(t) is strictly a function of time, and we likewise consider the forces F(t) to which this particle is subjected as strictly a function of time, then the behavior of this particle can be expressed in the form of total differential equations, because there is just a single independent variable, namely the time coordinate. Every physically meaningful variable exists as one of a countable number of explicit functions of time, and each of the values is realized at it’s respective time. Thus the total derivatives are evaluated over actualized values of the variables. In contrast, the partial derivatives over immaterial fields are inherently hypothetical, because they represent the variations in some variable of a particle not as a function of time along the particle’s actual path, but transversely to the particle’s path. For example, rather than asking how the force experienced by a particle changes over time, we ask how the force would change if at this instant of time the particle was in a slightly different position. Such hypotheticals have meaning only assuming an element of contingency in events, i.e., only if we assume the paths of material objects could be different than they are. 

If we were to postulate a substantial continuous field, we could have nonhypothetical partial derivatives, which would simply express the facts implicit in the total derivatives for each substantial part of the field. However, the intelligibility of a truly continuous extended substance is questionable, and we know of no examples of such a thing in nature. Given that the elementary force fields envisaged by the Maxwellians were eventually conceded to be immaterial, and their properties could only be inferred from the state variables of material entities, it’s clear that the partial derivatives over the field variables are not only hypothetical, but entail the assumption of freedom of action. In the absence of freedom, any hypothetical transverse variations in a field (i.e., transverse to the actual paths of material entities) would be meaningless. Only actual variations in the state variables of material entities would have meaning. Thus the contrast between total and partial differential equations reflects two fundamentally different conceptual frameworks, the former based on determinism and the latter based on the possibility of free acts. This is closely analogous to Aristotle’s dichotomy between natural and violent motions. 

As noted above, Einstein regarded this dualism as unnatural, and his intuition led him to expect that the field concept, governed by partial differential equations, would ultimately prove to be sufficient for a complete description of phenomena. In the same essay mentioned above he wrote 

What appears certain to me, however, is that, in the foundations of any consistent field theory, there should not be, in addition to the concept of the field, any concept concerning particles. The whole theory must be based solely on partial differential equations and their singularityfree solutions. 

It may seem ironic that he took this view, considering that Einstein is often regarded as a proponent of strict causality and determinism, but by this time he was deeply invested in the concept of a continuous field as the ultimate ontological entity, more fundamental even than matter, and possessing a kind of relativistic substantiality, subject to deterministic laws. In a sense, he seems to have come to believe that the field was not a hypothetical entity inferred from the observed behavior of material bodies, but rather that material bodies were hypothetical entities inferred from the observed behavior of fields. Part of this program was to eliminate the classical concept of forces acting between bodies, and to replace this with a fieldtheoretic model. He arguably accomplished this for gravitation with the general theory of relativity, which dispenses with the concept of a (violent) "force of gravity", at least for the passive response of test particles, and instead interprets the motions of objects under the influence of gravity as simply proceeding unforced along the most natural (geodesic) paths. Thus the concept of gravitational force, which was so central to Newton's synthesis, was fundamentally altered and reinterpreted. 

However, the concept of force is still very important in physics, partly because the active gravitational interaction between two masses still entails a mutual exchange of momentum which can be interpreted as a “force”, and also because it has not proven possible (despite the best efforts of Einstein and others) to do for the other forces of nature what general relativity did for passive gravitational motion, i.e., to express the apparently forced (violent) motions as natural paths through a modified geometry of space and time. 
