Following the success of the 1911 Solvay Conference and the rapid dissemination of the proceedings, the emphasis of research shifted towards the understanding of the spectra of atoms and molecules. With the availability of precision spectroscopic techniques, the bewildering variety of spectral features of atoms and molecules became apparent. The efforts described in Sect. 1.6 indicate how regularities were found in the patterns of spectral lines, the culmination of these investigations being the discovery of the formula for the Balmer series and the various formulae to account for the principal, diffuse and sharp series of the lines in the spectra of sodium, potassium, magnesium, calcium and zinc. As Planck remarked in 1902,

The Zeeman effect: Lorentz and Larmor's interpretations

In 1862, Faraday attempted to measure the change in wavelength of spectral lines when the source of the lines was placed in a strong magnetic field, but failed to observe any positive effect (Jones, 1870). Inspired by this negative result, Pieter Zeeman repeated the experiment and discovered the broadening of the D lines of sodium when a sodium flame was placed between the poles of a strong electromagnet (Zeeman, 1896a).

We have reached the goal I set myself when this journey began. From about 1927 onwards, the quantum theory in its modern guise for non-relativistic quantum mechanics was essentially complete, although there remained problems of interpretation which took a number of years to unravel – some of them are still hotly debated. But much of the apparatus was already in place and the subsequent developments changed completely the face of physics. Jammer (1989) summarises the achievement as follows:

As noted by Mehra and Rechenberg (2001), the 1930s also saw the beginning of the compartmentalisation of physics into separate quantum disciplines. Thus, during the 1930s,with the general acceptance and success of quantum mechanics, the quantum physicists began to specialise in disciplines such as atomic physics, molecular physics, solid state physics, including metal and semiconductor physics, condensed matter physics and low temperature physics, while at high energies, nuclear, particle and cosmic ray physics developed as disciplines in their own right. Whereas the pioneers of quantum mechanics regarded the whole province of quantum physics as their domain, the various branches of quantum physics became fragmented into these specialisms, not so different from those encountered in any physics department today.

The key role of experimental technique

1895 was not an arbitrary choice as the cut-off year for the history recounted in Chap. 1. Over the following years a number of experimental discoveries were made which were to change dramatically the face of physics. The origins of these discoveries can be traced to the need for increased precision in experimental physics. The industrial revolution and the widespread availability of electricity and electrical communication required more exact understanding of the physical properties of materials and also the establishment of international standards. These necessitated a much more professional approach to the teaching of experimental physics and its associated theory. As part of that movement the Clarendon Laboratory was founded in Oxford in 1868 and the Cavendish Laboratory in Cambridge in 1874. Of particular significance for this chapter was the foundation of the Physikalisch-Technische Reichsanstaldt in Berlin in 1887 with the task of providing precise measurements of the physical properties of materials which would be of importance to industry. There was an expectation that these laboratories would develop new techniques for undertaking precision measurements.

New experimental techniques were developed thanks to the development of new technologies. To mention only a few of the more important of these, better vacuums were available to physicists through the invention of the Geissler pump, invented by Johann Heinrich Wilhelm Geissler, a brilliant inventor and glass-blower, in about 1855. The vacuum was produced by trapping air in a mercury column and forcing it down the column by the force of gravity.

Optical spectroscopy, multiplets and the splitting of spectral lines

The achievements described in Chap. 6 represented a remarkable advance in the understanding of quantum phenomena, but there remained major challenges which were ultimately to undermine the successes of the old quantum theory. Continuing advances in spectroscopy enabled high resolution spectra to be obtained and, with the ability to place the sources of emission in strong electric and magnetic fields, the full complexity of atomic and molecular spectra became apparent. Atomic spectra display regularities, for example, the series spectra of elements such as sodium and calcium which could be described by the Rydberg formula (Sect. 1.6). Some of the most prominent spectral features consisted, however, of multiplets, meaning the splitting of a line into a number of separate lines with similar wavelengths. Examples of multiplets are illustrated in Fig. 7.1, derived from observations of the photosphere of the Sun from the Pic du Midi observatory.

The simplest lines are singlets, the example of the Hα line of the Balmer series of hydrogen being shown in Fig. 7.1a. In fact, the line is a very narrow doublet, which Sommerfeld attributed to the effects of special relativity upon the circular and elliptical orbits of electrons of the same principal quantum number (Sect. 5.3). The splittings we are interested in here are very much larger effects. The classic example of a doublet is the splitting of the sodium D line into two bright components labelled D1 (589.592 nm) and D2 (588.995 nm) (Fig. 7.1b).

Einstein in 1905

The next great steps were taken by Albert Einstein and it is no exaggeration to state that he was the first person to appreciate the full significance of quantisation and the reality of quanta. He showed that these are fundamental aspects of all physical phenomena, rather than just a ‘formal device’ for accounting for the Planck distribution. From 1905 onwards, he never deviated from his belief in the reality of quanta – it was some considerable time before the great figures of the day conceded that Einstein was indeed correct.

Einstein completed what we would now call his undergraduate studies in August 1900. Between 1902 and 1904, he wrote three papers on the foundations of Boltzmann's statistical mechanics. In 1905, Einstein was 26 and employed as ‘technical expert, third class’ at the Swiss patent office in Bern. In that year, he completed his doctoral dissertation on A new determination of molecular dimensions, which he presented to the University of Zurich on 20 July 1905. In the same year, he published three papers which are among the greatest classics in the literature of physics. Any one of them would have ensured that his name remained a permanent fixture in the scientific literature. These papers are:

1. (1) On a heuristic point of view concerning the production and transformation of light (Einstein, 1905a);

2. (2) On the motion of small particles suspended in stationary liquids required by the molecular-kinetic theory of heat (Einstein, 1905b);

3. (3) On the electrodynamics of moving bodies (Einstein, 1905c).

Copernicus

The Copernican heliocentric system

The great breakthrough that ultimately led to modern astronomy and cosmology came with Copernicus’ heliocentric system for describing the heavenly motions. But practically nothing one can say about Copernicus, either about his system or about its origins, is simple.

For one thing, the Copernican Universe is not the modern one, but the ancient cosmology in many of its most important respects. It is still a finite world bounded by a sphere of fixed stars. For another, Copernicus was not started on his path to his great discoveries by seeking a system that could use the Earth's motion to provide a more unified account of the apparent motions of the Sun and planets. Rather, he was dismayed by the appearance of the device of the equant point in Ptolemy's system and sought a means of eliminating it from an account of the heavens. Copernicus was even more devoted than the Greeks to the view that all the heavenly motions must be described in terms of pure circular motions in which everything moved uniformly about the circle with respect to the actual central point of the circle. Of course, hierarchies of circles, for example epicycles whose centers moved uniformly on deferent circles, were permissible.

The growth of theories

A very naïve view of science might go something like this: Scientists encounter a range of observable phenomena for which they have no explanatory account. Hypotheses are generated from the imaginations of the scientists who seek to explain the phenomena in question. These hypotheses are tested against the experimental results. If they fail to successfully account for those results, the hypotheses are rejected as unsatisfactory. But if they succeed in predicting and explaining that which is observed, they are accepted into the corpus of scientific belief. Then scientific attention is turned to some new domain of, as yet, unexplained phenomena.

This simple-minded picture of science has been challenged for a variety of reasons. Some are skeptical regarding the possibility of characterizing theory-independent realms of observational data against which hypotheses are to be tested. Others have noted the way in which the testing of hypothesis by data is a subtle matter indeed. It has often been noted, for example, that even our best, most widely accepted fundamental theories often survive despite the existence of “anomalies,” observational results that are seemingly incompatible with the predictions of the theories.

The background to the Principia

In Newton's great work dynamics is presented as being derivable in a systematic way from a small number of fundamental first principles. In this Newton resembles Descartes. But, unlike Descartes, the first principles are not alleged to be derivable a priori from “clear and distinct ideas.” They are, rather, painstakingly inferred from the known lower-level generalizations, themselves inferred from observation and experiment, as the best basic principles from which the known phenomena can be derived. And, very much unlike Descartes, and very much in the tradition of Galileo and Huyghens, Newton is extraordinary in his ability to apply the methods of mathematics to the description of particular dynamical situations so that detailed and exact characterizations of the situation can be formulated, and precise predictions made.

Newton's Principia appeared only after many years of the careful exploration of dynamics and its applicability to a theory of the world by its author. Most of this earlier work remained unpublished. There seem to be several reasons for this, including many forays by Newton into other fields, such as the invention of the calculus, brilliant experiments on light, including the discovery of the dispersion of white light into colors and basic interference phenomena, and such matters as the invention of the reflecting telescope. Newton's sensitivity to what he took to be inappropriate criticism of early public work may have also contributed to a reluctance on his part to publish.

Some Greek knowledge and speculation

The motion of the heavenly bodies, observed as lights in the sky, provides us with a remarkable spectacle of a phenomenon describable in a small number of terms and exhibiting an easily noted regularity in space and time. This spectacle caught the attention of many cultures in the beginnings of their attempts to characterize the world as a place of some describable order. Most of the cultures were unable to get beyond the ability to discover numerical formulae that allowed one to predict recurrences in the domain of the heavens, sometimes with astonishing accuracy. In ancient Greece, however, astronomy took science further. In particular, Greek astronomy involved deep connections with Greek attempts to construct a general dynamical theory of motion and its causes. This close connection between dynamics and astronomy persisted throughout the history of classical dynamics, as we shall see. It is necessary for us, therefore, to say a little bit about some of the aspects of Greek astronomy that impinged upon Greek theories of motion.

By the time of the Greek classical era, many important facts were well known and widely agreed to. Whereas early Greek speculation about the shape of the Earth thought of it as flat, perhaps a disk of land surrounded by a circumventing ocean, it was soon an accepted fact that the Earth had the form of a sphere. Observations on how the elevations of stars changed as one moved north or south, how ships disappeared a little at a time over the horizon and how the length of a day varied with latitude could be explained only by invoking such a shape for the Earth. By analogy, models of the Moon and Sun as illuminated disks were soon replaced by accounts of these heavenly bodies as also spherical. (It is remarkable how spherical the Moon looks, in fact, when seen during a total eclipse.)

From special problems and ad-hoc methods to general theory

Our outline of the development of mechanics has proceeded as if we could tell a story with a single line of development. But our account up to now has been, as we shall see, somewhat misleading. Even prior to the great Newtonian synthesis, other approaches to the solution of the problems of dynamics were simultaneously being explored. Most of these approaches remained fragmentary and partial until the eighteenth century. For that reason we have neglected them, reserving discussion of them until the more extended discussion of how those programs became solidified, generalized and systematized in the later history of dynamics.

At this point, however, it becomes impossible to deal with matters in a strictly chronological manner. In the years following the publication of Newton's Principia dynamics followed a number of distinct, although deeply related, patterns of development. It will be essential to deal with each of these in turn, forcing us to go over the same temporal period from several perspectives. This chapter is preliminary to those that follow. In it I will try to lay out something of the problem situation facing the great developers of mechanics and outline the basic structure of the multiple approaches suggested to deal with that array of problems. We may then proceed to explore the several approaches in detail one at a time.

From least time to least action

The third developmental stream of dynamics that came to fruition in the eighteenth century is quite unlike the first two streams in significant ways. It did not arise out of repeated attempts at solving particular difficult special cases of dynamical problems. Nor, when it was discovered, was its primary importance its ability to provide new methods for solving such problems. Its importance for the discipline was, rather, more of a “theoretical” kind, providing new and deep insights into the fundamental structure of the theory. Whereas the other two streams of development carried with them philosophical issues already present in the standard controversies over the mode of explanation in dynamics familiar to Newton and his critics, this third developmental stream opened up entirely new controversial issues concerning what could count as a legitimate explanation in science. It was also curious that these developments in dynamics had their origins not in contemplation of mechanical issues, but, rather, in explanatory accounts offered in the theory of light, in optics, dating back to ancient Greece.

The Greek mathematicians had become aware of “minimization” problems quite early. For example, it was well known that the circle was the shortest curve bounding an area of specified size. Indeed, this is one of the “perfections” of the circle taken by Aristotle to account for the cosmic orbits. The use of a minimization principle by the ancient Greeks that eventually fed into the important role such principles play in physical explanations, though, was the proof by Hero that the law of reflection, namely the angle of incidence of the light on the surface of the mirror equaling the angle of reflection, could be derived by postulating that the distance taken by a light beam to go from a point to a point on the mirror and then to a third point of reception was least when the mirror point was such that the distance traveled by the light along the path was less than that which would be traveled for any other point on the mirror taken as intermediary. The proof is from plane geometry and is very simple.

Newton's masterful achievement was constructed under the influence of much previous philosophical discussion and controversy that went beyond the limits of scientific debate narrowly construed. Much that Newton says in the Principia also ranges beyond the confines of experimental, or even theoretical, science and passes into the realm of what we usually think of as philosophy. Newton's work gave rise, possibly more than any other work of science past or future, excepting just possibly the work of Darwin and Einstein, to vigorous philosophical as well as scientific discussion. Let us look at some of the philosophical issues behind, within and ensuing from Newton's work.

It is convenient to group the discussions into three broad categories. First, there is the “metaphysical” debate over the nature of space, time and motion. Next there is the debate over what can be properly construed as a scientific explanation of some phenomenon. Lastly, there is the controversy over what the appropriate rules are by which scientific hypotheses are to be credited with having reasonable warrant for our belief. We will discuss these three broad topics in turn.

Our concern in this book is with dynamics, namely the science of motion, its description and its causes. But mechanics traditionally had two branches, dynamics and statics, the latter being the science of the unmoving, that is of how forces can jointly result in unchanging, equilibrium, states of systems. Although we shall deal with statics only in a brisk and cursory manner, we must pay some attention to its historical development, since principles developed in statics played a fundamental role in the foundations of some important approaches to general dynamical theories. Before moving on to the development of dynamics beyond Newton's great synthesis, then, we shall have to spend at least a little time surveying some aspects of the development of statics prior to the eighteenth century.

Two basic areas of investigation constituted the initial exploration of statics in ancient Greece. The major set of problems that gave rise to statics consisted in attempts at describing the general laws governing the equilibrium conditions for simple machines. In particular, the lever and the inclined plane were the characteristic problems tackled. A second branch of statics began with considerations of the static behavior of objects immersed in fluids, consideration of which constituted the first efforts at understanding the statics of fluids, hydrostatics. It is the former problem area, though, that is of most interest to us.

Throughout this book so far our attention has been firmly directed to issues arising out of consideration of the fundamental postulates of dynamical theory. What should those posits be? How are the alternative choices for foundational axioms related to one another? What are the basic concepts utilized by those posits and how are they to be interpreted? What is the epistemic status of the basic postulates? And, finally, what kind of a world, metaphysically speaking, do the posits demand?

In this chapter, however, we will look not at the fundamental posits but, instead, at how the consequences of these posits are developed in particular applications of dynamical theory. We have generally been avoiding issues of application. But some of these realms of application are quite fascinating from an historical point of view, for it is often an extremely difficult task for the scientific community to figure out how the elegant basic posits of the theory are actually to be used in describing the complex systems we find in nature. Great efforts in classical dynamics were devoted to such issues as the appropriate frameworks for applying the theory to the motion of rigid bodies, to fluids of ever more complex nature, going from the non-viscous to the viscous and from the incompressible to the compressible, and to more complex continuous media. Difficult special cases, such as those of shock waves, where familiar continuity assumptions cannot be maintained, also led to the development of subtle and sophisticated application programs for the dynamical theory.

The variety of explanatory modes in dynamics

We have seen that dynamics is a theory with a multiplicity of explanatory structures that can be used to formulate its lawlike conclusions and to provide explanations of the behavior of systems within its purview. We have also seen that the threads of some of these structures can be traced back to the earliest days of the development of the theory.

One pattern of explanation in dynamics we might call the “Newtonian.” Here one must first posit an appropriate structure that admits a preferred metric of time and the existence of the preferred inertial reference frames to which all motion is to be referred. Inertial motion with constant speed and direction is taken as the “natural,” “unforced,” state of a body.

Inertial mass and force are introduced. The former is an intrinsic property of a piece of matter representing its resistance to having its state of motion changed, and the latter is the (vector) measure of the influences that can change the state of motion of a system. The fundamental law, of course, is the proportionality of the linear momentum change of the system to the force applied to it. This initial Newtonian framework must be supplemented, as was first seen by Euler, by a corresponding notion of moment of inertia as intrinsic resistance to change of state of rotation and of moments of forces (or torques) as the measure of the influences generating changes of angular momentum.