The theory this book is exploring, Newtonian dynamics, is, of course, a false theory of the world. Several scientific revolutions have shown us that it must be replaced as our fundamental theory of motion and its causes.

The special theory of relativity rejects even the basic kinematics of the Newtonian account. Even at the level of the very description of motion, at the level of our construal of the spatial and temporal structure of the world, the Newtonian account is rejected. At the level of dynamics, the level at which the causes of changes of motion are explored, the theory, once again, replaces the Newtonian account with a novel theoretical structure. The general theory of relativity proposes even further, highly dramatic, changes in our very notions of space and time. Its most direct contact with the Newtonian picture of the world is to replace Newton's famous account of the origin, nature and effects of gravitational force with a new account of the gravitational interaction of the material of the world.

Many threads were finally woven together in the great Newtonian synthesis from which all further developments in classical dynamics followed. In this chapter we will outline a few of the major contributions to dynamics that followed the Copernican revolution in astronomy, but preceded the final accomplishment of a full theory that we can recognize as classical dynamics. Three names dominate this early work on dynamics, those of Galileo, Descartes and Huyghens, but, as we shall see, important contributions were made by less well-known figures as well.

Galileo

Almost everyone would give Galileo credit for initiating the great revolution in dynamical theory that ultimately led to Newtonian classical dynamics. But, as is the case in the work of many originators of a new science, there is no simple way of characterizing Galileo's contribution. His work, while being innovative in a revolutionary way, retains much that in retrospect seems quite conservative in its nature, borrowing much in the way of concepts, views and arguments from his predecessors. His exposition is sometimes quite informal, presented in the form of a charming dialogue in some cases. This sometimes makes it rather difficult to say exactly what Galileo believed, since, in some crucial cases, he seems to hold a number of distinct opinions simultaneously. In some crucial cases we would now be inclined to think that Galileo got things very wrong indeed. But even in his errors he provides deep enlightenment.

The realization that a branch of knowledge could be presented in a form in which the entire contents of the field of investigation could be expressed by positing a small number of basic truths and by claiming that all the other truths of the discipline followed from these basic posits by pure deductive reasoning alone predated any serious development of dynamics or of other branches of physical science. The axiomatization of geometry has its origin at such an early date, in fact, that we have no good record of when or how the very idea of presenting geometry as a deductive formal discipline arose.

This early discovery of a branch of mathematics as a formal science had many consequences for the history of science and the history of philosophy of science. The entire history of the rationalist approach to knowledge in philosophy is founded on the early discovery that geometry could be structured as a set of consequences logically deducible from apparently “self-evident” first principles. Closer to our concerns, it is clear that Newton's Principia is itself structured to resemble as closely as possible the standard presentation of geometry. But there is, of course, no pretence on Newton's part that his first principles could themselves be established without reference to empirical experiment. (It was left to Kant to fall into the trap of trying to establish Newtonian dynamics as a fully a-priori science!)

The nineteenth-century reconstructions of dynamics we have just surveyed were certainly not motivated by philosophical reflections on the traditional foundations of the theory. Nor were they formulated in response to any felt disquietude about the traditional theory. Hamilton's equations arise out of an understanding, developing at the time, that a single second-degree differential equation could be replaced by a pair of coupled first-degree equations. Hamilton–Jacobi theory has a less purely mathematical motivation, in that it followed from a deep understanding of the degree to which the formalisms of geometrical optics and of dynamics bore interesting parallels to one another. Here the primary inspiration comes from the understanding that the Principle of Least Time in the former theory and the Principle of Least Action in the latter were sufficiently similar that other formal similarities, such as advancing wave fronts and trajectories as rays orthogonal to these, might be found as well. The bracket formulation of dynamics is, once again, a purely formal manipulation of the theory.

Each of these reconstructions, as we have seen, has manifold consequences. They do play some role in extending the ability of the theory to be applied to difficult special cases. But they also provide deeper understandings of the hidden internal structures of the theory, as in, for example, the realization of the fascinating algebraic structure among the generalized variables revealed by the bracket notation or the “wave-front” structure in configuration space revealed by Hamilton–Jacobi theory. And they also provide just the resources that will later be needed in going beyond classical dynamics to newer theories, as in the application of Hamilton–Jacobi theory to the foundations of the wave-theory version of quantum mechanics and the application of the bracket notations in the formalizing of the matrix version of quantum theory.

From virtual work to “d'alembert's principle”

The Newtonian approach to dynamics had its origins in Newton's great work. It is by the full generalization of the Second and Third Laws to make them applicable to all parts of any complex system, including infinitesimal parts, and by adding to the linear laws those appropriate to rotation that the full theory is obtained. The driving force behind the discovery of the full methodology was the ongoing program of finding solutions to particular difficult problems of statics and dynamics. It was only by coming to grips with such issues as the shape of a hanging chain, the vibrations of a drumhead, the motion of a rotating rigid body, and the dynamics of fluid flow that the general principles became apparent.

The developmental stream we are now about to explore also has its origins in the attempt to solve particular difficult problem cases in dynamics. But it develops not out of Newton's work, but out of the methods of statics that long predate the Principia. The problems attacked are those involving constrained motion. In the rotation of a rigid object, one might think of each point mass making up the object as constrained to maintain a fixed distance from each other point mass making up the rigid body. Or one might try to determine the dynamics of a body confined to some geometric figure, such as a bead constrained to slide on a rigid rod of some shape when some motion is applied to that rod. Or, perhaps, one might be dealing with a wheel or a ball constrained to roll frictionlessly on a plane surface. The key to solving these problems is to find some method by which the forces of constraint need not themselves be calculated. As we shall see, the method developed implicitly goes beyond Newton in the same way as the improved Newtonian approach did, in that the roles of torques and angular momenta are taken into account along with the forces and the linear momenta.

Kinematics, dynamics and constitutive equations

The development of the Newtonian approach required the sorting out of several distinct components needed for dynamically characterizing a system. In many of the approaches to specific problems, all of these constituent elements were used. But it took quite a while for the realization to sink in that a clear understanding of a general method for solving dynamical problems required the separation of the elements into components that presented themselves individually in the overt method of solution.

One component was kinematics. This encompasses the general principles for simply describing the motions of systems. Here the obvious fundamental concepts are those of space and time. Position, velocity and acceleration, as well as the notion of a time interval itself, are fundamental concepts. Geometry, and its generalization to take account of the element of time added to those of space, is the fundamental discipline needed.

One of the earliest great breakthroughs was the application of coordinate methods and algebraic manipulations of those coordinates to describe the kinematics of systems. Here Fermat and Descartes were the great innovators. We have already noted the advances achieved in the early eighteenth century. These were the application of the methods of the calculus, originally due to Newton and Leibniz, to generalize the algebraic methods, and the realization that many problems could be more easily dealt with if the coordinates were all referred to a single orthogonal coordinate system fixed in one state of motion in space, rather than to relevant directions fixed by the components of the system to be described.

The Hamiltonian formalism

The nineteenth century saw several new proposals for fundamental principles to be placed at the foundations of dynamics. In 1829 Gauss offered the “Principle of Least Constraint.” Let a system of particles be connected together by constraints. Suppose each particle n starts at An and after a time interval ends up at Bn. Let Cn be the position it would have ended up at at the end of the time interval had the constraints not been imposed. Then, for the actual motion undergone, the sum of the mass of each particle times the square of the distance from Bn to Cn will be minimal over the class of all motions compatible with the constraints binding the particles together.

In 1894 Hertz offered the “Principle of Least Curvature.” Generalizations of the ordinary Euclidean notions of distance along a path and curvature of a path are constructed. It is then shown that, using these definitions, a system of particles that is isolated, that is not subject to some external force, will evolve in such a way that the motions of the individual particles will generate a path in a multi-dimensional space for the point representing the system as a whole that has, at each point, minimal curvature in the sense defined by Hertz. This principle provides a useful opening for applying methods of differential geometry in dynamics, for in a curved space the paths of least curvature, the geodesics, are objects of intensive study in differential geometry. Hertz's principle then tells us that, with the proper definitions of distance and curvature for our space, one can view the dynamical trajectory of a system as a geodesic in the representing space.

Machian dynamics

We have seen how Newton attacked spatio-temporal relationism not through a typical philosophical argument that might be argued by a substantivalist, but by claiming that “absolute” space and time were necessary for dynamics. Inertially moving systems are radically distinguished from those which, under the influence of external forces, are not moving inertially. But, to make sense of motion in a constant direction and at a constant speed, we need an absolute frame relative to which constant direction is constant and relative to which our changing positions are to be measured, and we need an absolute lapse of time (up to a linear transformation) relative to which constant speed is constant. The discovery that absolute place and velocity play no role and can be eliminated even in the framework of Newtonian dynamics in favor of Galilean or neo-Newtonian spacetime with its class of inertial frames should give the relationist no comfort.

In Chapter 16 we briefly sketched the version of relationism proposed by Mach. The core of this program sought a “cosmic” solution to the problem of the reference frame relative to which accelerations were absolute, with the source of the inertial forces sought in acceleration-dependent relations of test systems to the distant “fixed stars.” We noted that Mach's proposal remained shrouded in vagueness and that subsequent proposals in his vein were generally deemed unsatisfactory. In the twentieth century a new kind of Machianism has been proposed by J. Barbour and B. Bertotti. It is to their theory that we must now turn our attention.

At this point it will be useful to interrupt our exposition of the historical development of dynamics in order to make some brief retrospective observations on how the development of the theory up to the later part of the eighteenth century carried with it an ongoing and evolving debate about the very nature of scientific explanation and scientific theory. What are the legitimate forms an explanatory account can take in science? What are the legitimate concepts that may be employed in such explanations? What are the fundamental posits of our theory, and what are the legitimate grounds by which we may justify our beliefs in the fundamental posits of our scientific account of the world?

The Aristotelian account of dynamics, and Aristotle's related account of cosmology, employed explanatory notions adopted from our pre-scientific, “intuitive” ways of answering “Why?” questions about the world around us. The ultimate origin of our employment of such explanatory structures is a worthy topic for exploration, but one we will not be able to embark on here. The notion of efficient cause presumably comes from our everyday experience, primarily, one imagines, experience of things pushing and pulling each other around. This “everyday dynamics” as it appears in our daily experience may very well be the source of our idea that explanations are to be given in terms of something like Aristotelian efficient causes. The idea of an explanation given in terms of final causes may have its origin in the fact that so much of our activity as agents is accounted for by means–ends explanations in which motives, purposes and goals play such a significant role. Again the idea that various components of living beings, their organs, all have their specific functions or roles, and that their existence must somehow be accounted for in terms of those roles, long predates science properly so-called.

In this chapter we will explore three more stages in the nineteenth-century development of dynamics. One program was not itself directed at finding new foundational posits for the theory. Its initial purpose, rather, was to supply a method to facilitate the solving of dynamical problems, especially when they were framed in the mode of the Hamiltonian dynamical equations. We need to pursue it a bit, however, since it provides some of the resources needed to understand the second program treated in this chapter.

This second program is Hamilton–Jacobi theory. Here, starting from Hamilton's work in optics, rather than in dynamics, the result was the development of new possible foundational equations for dynamics to supplement those already known. Just as in the case of the development of the Hamilton dynamical equations, there was no claim here that the results went beyond the existing foundational posits in any fundamental way. It was universally accepted that the existing foundational methods were correct and complete as they stood. Rather, a new “reformulation” of the existing foundations was what was on offer.

Discovery, controversy and consolidation

As is the case with so many of the fundamental concepts of dynamics, Galileo provided the first seminal ideas. In his discussion of falling bodies constrained to inclined planes, he offered an ingenious argument to the effect that the speed obtained at the end of the fall will be dependent only upon the height of the fall, and not on the slope of the incline, at least insofar as friction and air resistance can be ignored.

The argument rests upon intuitive agreement that an object suspended by a string from a pivot point and allowed to fall from a certain height will return to the same height even if the string encounters a nail around which it must pivot at some point in the descent. Two deep notions are encountered here. One is that height in a gravitational field is the unique parameter needed in order to capture the potential for a fall from that height to be able to generate a specific quantity of motion. Here we have the beginnings of the notion of potential energy. The other idea is that the motion generated by a change in height will be independent of the path by which that loss of height is obtained. Here is the beginning of that essential aspect of potential energy that the change in potential energy on going from one point to another, and hence the quantity of motion generated or absorbed, is path-independent. Galileo is clearly aware of the essential difference between the energy “stored” by a gain of height and the energy “lost” owing to, say, friction of the inclined plane or air resistance. In the former case the quantity of motion that disappears can be made to reappear merely by restoring the object to its original height. In the latter it cannot. In the former case the motion temporarily lost is path-independent, depending only on the height gained. In the latter the motion that vanishes is path-dependent.

Classical dynamics has a very special place within our theoretical description of the world. For one thing, the theory has had a “lifetime” within physics that is nothing short of astonishing. Beginning with the earliest attempts at a characterization of motion and its causes within ancient Greek science, developing slowly but with some sureness through the Islamic and medieval European eras, exploding into a grand synthesis in the Scientific Revolution, and showing still further important development in its foundations and in applications from then to the present, the theory's place in science is one not of years or centuries but of millennia.

Even now, after having been displaced by relativistic theories and quantum theories and no longer being considered the central “truth” of theoretical physics, the theory still surprises us with new formulations, new applications and new interpretations. Theories in fundamental physics are typically formulated with a characterization of basic states and their possible configurations and changes that is formally similar to the kinematics of dynamics, and with a characterization of changes of these states over time by means analogous to the dynamical parts of classical dynamics. And in the case of relativistic and quantum dynamics the crucial role played by classical dynamics is even clearer since, contrary to some radical “revolutionary” views of the history of science, the important ancestral relationship of the classical dynamical concepts to those of the newer theory is clear.