The simplest motion occurs in one-dimensional systems subjected to time-independent forces. The most important characteristics of regular (non-chaotic) behaviour will be demonstrated by means of such motion, but this also provides an opportunity for us to formulate some general features. The overview starts with the investigation of the dynamics around unstable and stable equilibrium states, where the essentials already appear in a linear approximation. Outside of a small neighbourhood of the equilibrium state, however, non-linear behaviour is usually present, which manifests itself, for example, in the co-existence of several stable and unstable states, or in the emergence of such states as the parameters change. We monitor the motion in phase space and become acquainted with the geometrical structures characteristic of regular motion. The unstable states, and the curves emanated from such hyperbolic points, the stable and unstable manifolds, play the most important role since they form, so to say, the skeleton of all possible motion. In the presence of friction, trajectories converge to the attractors of the phase space. For regular motion, attractors are simple: equilibrium states and periodic oscillations, implying fixed point attractors and limit cycle attractors, respectively.

Instability and stability

Motion around an unstable state: the hyperbolic point

Let us start the analysis – contrary to the traditional approach – with the behaviour at and in the vicinity of an unstable equilibrium state.

An equilibrium state of a body at some position x* is unstable if, when released from a slightly displaced position, the body starts moving further away from x*.

A special but important class of dynamics is provided by systems in which friction is negligible, or, more generally, where dissipative effects play no role. In this case the direction of time is not specific, the process described by a differential equation is reversible: forward and backward time behaviour is similar. Think of, for example, a planet: one cannot decide whether its motion recorded on a film takes place in direct or in reversed time. In frictionless systems phase space volume is preserved, and attractors cannot exist. In such conservative systems, the manifestation of chaos is of a different nature than in dissipative cases. In this chapter we investigate persistent conservative chaos where escape is impossible, and defer the problem of transient conservative chaos to Chapter 8. We start with the area preserving baker map and the stroboscopic map of a kicked rotator. Next, the dynamics of continuous-time, non-driven frictionless systems is considered. On the basis of these examples, we summarize the general properties of conservative chaos, including one of the most important relationships, the KAM theorem. The structure of chaotic bands characteristic of conservative systems is discussed and compared with that of chaotic attractors. Finally, we present how conservative chaos of increasing strength manifests itself and we discuss the consequences.

What is a fractal?

Objects with large surfaces

It is taken for granted that the surface or volume of a traditional geometrical object, for example a sphere or a cube, is well defined. Indeed, filling the object with smaller and smaller cubes leads to better and better approximations, and the total volume of the cubes converges to that of the object in question. It is well known that the surface, S, is proportional to the second, while the volume V is proportional to the third, power of the linear size, L, of the object. Consequently, the surface-to-volume ratio, S/V, is proportional to V−⅓. (For plane figures, the ratio of the perimeter, P, to the area, A, is proportional to A−½.) The surface-to-volume ratio is therefore finite and becomes smaller as the size becomes larger. This is why surface phenomena are of little importance compared with volume phenomena for macroscopic systems of traditional geometry.

On the other hand, it is known that there exist macroscopic objects with large surface area. These are always porous, with ramified or pitted surfaces. Effective chemical catalysts, for example, must have a large surface. The need for rapid gas exchange accounts for the large surface-to-volume ratio of the respiratory organs. The surface area of the human lungs (measured at microscopic resolution), for example, is the same as that of a tennis court (approximately 100 m2), while the volume is only a few litres (10−3 m3).

The world around us is full of phenomena that seem irregular and random in both space and time. Exploring the origin of these phenomena is usually a hopeless task due to the large number of elements involved; therefore one settles for the consideration of the process as noise. A significant scientific discovery made over the past few decades has been that phenomena complicated in time can occur in simple systems, and are in fact quite common. In such chaotic cases the origin of the random-like behaviour is shown to be the strong and non-linear interaction of the few components. This is particularly surprising since these are systems whose future can be deduced from the knowledge of physical laws and the current state, in principle, with arbitrary accuracy. Our contemplation of nature should be reconsidered in view of the fact that such deterministic systems can exhibit random-like behaviour.

Under certain circumstances chaotic behaviour is of finite duration only, i.e. the complexity and unpredictability of the motion can be observed over a finite time interval. Nevertheless, there also exists in these cases a set in phase space responsible for chaos, which is, however, non-attracting. This set is again a well defined fractal, although it is more rarefied than chaotic attractors. This type of chaos is called transient chaos, and the underlying non-attracting set in invertible systems is a chaotic saddle. The concept of transient chaos is more general than that of permanent chaos studied so far, and knowledge of it is essential for a proper interpretation of several chaos-related phenomena. The basic new feature here is the finite lifetime of chaos.

In dissipative systems transient chaos appears primarily in the dynamics of approaching the attractor(s). It is therefore also called the chaotic transient. The temporal duration of the chaotic behaviour varies even within a given system, depending on the initial conditions (see Fig. 6.1 and Section 1.2.2). Despite the significant differences in the individual lifetimes, an average lifetime can be defined. To this end, it is helpful to consider several types of motion (trajectories) instead of a single one: the study of particle ensembles is even more important in transient than in permanent chaos.

1. Dingler refers to the well-known example of the accelerating railway car. Although by now this example has become rather trivial as a result of its frequent discussion and was completely clarified by relativity theorists, particularly in connection with Lenard's repeatedly proposed misunderstanding, we will consider it again here for the sake of completeness. Dingler explains that the braking railway car would not be comparable to one in uniform motion because in the first case the (negative) acceleration does not directly act on the objects in the car, but rather will be first transferred to them by elastic forces. Only in a “real” gravitational field, e.g., one that is parallel to the rails and brought about by a single large mass, can we have complete equivalence; but here again there is no perceptible difference between uniform and accelerated motion if we discount frictional forces. The latter is true, but the standpoint of Newtonian theory does not provide an adequate explanation of it because the accelerated train in this example represents a system (in free fall), for which the Newtonian laws apply, although it itself experiences an acceleration relative to the collection of uniformly moving inertial systems of the cosmos.

2. […]

A clarification of the fundamental concepts of the theory of relativity will only be possible if the tenets of the theory are presented in axiomatic form. The axioms contain the fundamental facts whose existence justifies the theory; in principle, they are empirical assertions capable of experimental verification. In addition to these are the definitions through which the theory's conceptual content is constructed. These, in contrast to the axioms, are arbitrary forms of thought, capable of neither empirical confirmation nor refutation. Their arbitrariness is only constrained by certain well-understood logical demands that are required for a scientifically appropriate system. They must be univocal; but moreover they must lead to a scientific system characterized by certain properties of simplicity. Whether they fulfill these demands is not solely a matter of form, but depends upon the validity of the axioms; the preferred properties of the conceptual system derive from the facts laid down in the axioms. The results of the systematic derivation of all the theorems of the theory from the definitions and axioms, viz., “ordine geometrico,” are twofold: on the one hand the discovery of those facts without which the theory could not exist and whose validity completely proves the theory, and the other hand the discovery of its logical structure which gives the theory its legitimate place as a scientific theory.

The axioms fall into two classes: light axioms (I–V) and matter axioms (VI–X).

Having already given a comprehensive account of an axiomatization of the theory of relativity, I would now like to expand upon those consequences which are especially important for physics. While this was originally intended to be a work of epistemology, and its real significance is primarily in that field, it does, however, provide some important results for the experimental foundation of the theory of relativity.

There are two approaches to physical axiomatization. In formulating a deductive axiomatization, the most general principle possible, perhaps a variational principle, is placed at the top and all other details are derived from it; only these derived details are testable, and if they are verified we may consider the abstract axiom to be more or less probable. Constructive axiomatization, on the other hand, proceeds differently. Here we take as axioms only those statements that are themselves direct experimental results; from them we derive the entire theory by integrating some additional conceptual elements, viz., definitions. The definitions are arbitrary and therefore can never introduce error into the theory. This form of axiomatization has the great advantage for physics that the implications of each experimental result can be immediately recognized.

Fifty years after the untimely death of Hans Reichenbach, the evolution of his views on space, time, and motion is receiving much due critical attention. While his mature views, expressed most famously in his book Philosophy of Space and Time, have long been commented upon, challenged, and amended by some of the most important figures in contemporary discussions of the philosophical foundations of physics, only in the last decade or so – a couple of generations removed from the heyday of logical empiricism – is the work of Reichenbach and contemporaries like Moritz Schlick and Rudolf Carnap being widely considered in terms of its value to the history of ideas.

This collection seeks to contribute to that growing conversation by bringing together English translations of nine essays from 1920–25, the period preceding Philosophy of Space and Time, that have not appeared in earlier collections of Reichenbach's writings. These articles range from technical discussions published in scientific journals, to overtly philosophical discussions and responses to philosophical opponents published in philosophical journals, to semipopular pieces designed to set out Reichenbach's interpretation of relativity theory in clear, explicit terms. It is hoped that by providing access to additional “data points,” the discussion of the emergence of Reichenbach's later views may be advanced.

The first half of the 1920s was a period of crucial importance both to the burgeoning movement that would become analytic philosophy and to the philosophical development of Hans Reichenbach personally.

1. The philosophical literature today is moving off in two divergent directions. Both share the goal of understanding the foundational facts underlying the knowledge process and which can only be experienced individually; but while one direction strives to illuminate and clarify these facts by breaking down the terms in which the facts are expressed, it suits the needs of the other to obscure them. The latter, of course, is always possible since the words, which surround the facts like a mushy porridge, are difficult to control and in their emptiness are often completely irrefutable. It should therefore be stated from the start that Schlick's representation belongs to the first direction, and this must be considered its initial virtue in a time of increasing philosophical unrefinement (including among the tenured philosophers).

2. Its second merit is that it bases its critique upon a systematic construction including a closed framework of concepts, and does not at all shy away from manipulating and shaping them until they are integrated into this framework. This is an act deserving of thanks because it creates a work of thought that gives rise to discussions whose cognitive value can be clearly assessed. In addition, the language used is pleasingly sober, completely excluding all affected attitudes and not confusing ethical sentiment with epistemological clarification.

3. […]

It is strange that we wait to subject a science's fundamental concepts to a pointed critique until it has reached a certain degree of completeness. So it is that the concepts of space and time have only now found a final clarification through Einstein's theory of relativity after physics was already a mostly complete science. But we can now say that it deserves the glory of a real, exact, natural science since the hitherto unnoticeable veil has been lifted from its elementary concepts.

The concept of motion as a change in position over time combines the most important categories in physics; for that reason its investigation is the starting point for the solution to space-time problems.

The most simple conception of motion is the “kinematic.” For example, if a billiard ball moves on a horizontal surface, its position is characterized by its distance from two adjacent edges of the billiard table; that the ball is in motion is a function of these changing distances. We speak of the kinematic concept of motion because it consists only of changes in spatial lengths. But what would one observe if the billiard ball was portrayed as a fixed point and the billiard table underneath was moved? Again, there are changes in both distances. Assume that I see the billiard table through a cardboard tube, so that I can only survey the green surface.

1. This view allows for speeds whose numerical value is greater than the speed of light. This, however, does not contradict the theory of relativity, since the number 3.1010 cm sec–1 represents an upper limit only for inertial systems. Superluminal speeds, in the strict sense of the word, are non-existent, since no body can be moved quicker than a light signal at the same location in space and time in an inertial system. Further, in a gravitational field, light is the fastest messenger. The numerical value of this speed is dependent upon the definition of time in a gravitational field. Hence, in a gravitational field one cannot simply apply the Lorentz contraction of the special theory of relativity. In no way does it follow from Einstein's formulae that the planets would be shortened in the direction of their motion relative to a coordinate system attached to the Earth.

2. That the fixed stars, relative to a stationary Earth, all have the same 24-hour period of revolution is also no coincidence on this view.

3. […]

A comprehensive work that sets out the philosophical problems of the theory of relativity in detail would be a true enrichment of the literature. Unfortunately, Müller's volume contains mistakes. It distinguishes itself by committing these errors in a very dull and matter-of-fact way and not in the usual emotion-laden warlike fashion of most of Einstein's opponents – yet it is built on an entirely flawed understanding of the special theory of relativity that results in faulty conclusions. A necessary prerequisite for any philosophical critique of relativity is an analysis of the theory's factual basis and conceptual foundation; and Müller's efforts are frustrated by the inability to offer such an analysis. His first mistake concerns the concept of simultaneity. He claims that simultaneity can be known and does not need to be defined. He fails to grasp that there are two types of definitions: the definition of concepts within a conceptual system and “coordinative definitions” which specify how given concepts acquire an empirical reality. Thereby, a conceptual definition is posited for a unit length; but in order to actually carry out measurements a coordinative definition must be put in place to fix that “this rod here” is a unit long.

In a recently published article, I have reported on an axiomatization of the relativistic space-time theory. In light of this, we may now test the possibility of absolute time by investigating which axioms are and are not compatible with it.

One possibility for defining the synchronization of clocks, so that the same simultaneity relation holds for all systems, arises from the transportation of clocks. Two clocks that are brought into synchronization at the same place are to be called synchronized as well when one or the other is transported to a different place. This is a definition of simultaneity and is neither true nor false, but an arbitrary rule. For that reason it can be used in any case; but in order to be univocal, it must satisfy the following axiom:

Axiom A

Two clocks that are synchronized at one place are always synchronized when compared at the same place regardless of the paths of transport.

By “clock” we understand here a closed periodic system. is satisfied is a mere matter of fact and can be decided independently of the definition of simultaneity for distant places and independently of the theory of relativity. The theory maintains ‘and it has received extremely indirect confirmation’ that axiom A is false, and therefore it rejects absolute time; but to the present, no means had been found to directly test the axiom.

Recently, Hj. Mellin, in a lengthy examination, offered a critique of my Axiomatization of the Theory of Relativity. A discussion of the objections that Mellin raises to my axiomatization, and thereby to the theory of relativity, seem to me to serve the general interest because of their fundamental nature and his clear formulation of views which are most often only operative on a subconscious level, and I would therefore like to answer them here.

1. The most significant difference in our positions lies in our understandings of the relationship between the mathematical discipline of geometry and reality. Here, I adopt the perspective (which is often incorrectly termed conventionalism) that the geometrical axioms as mathematical propositions are not at all descriptive of reality; this only occurs when physical things are shown to be coordinated to the elements of geometry (coordinative definitions). If we take very small bits of mass to be points, light rays to be straight lines, and the length of a segment to be determined by the repeated placement of a rigid rod, then the statement that straight lines are the shortest becomes an (empirically proven) statement about real things. Without such coordinative definitions, these propositions say nothing about reality. Mellin's objections to this view rest on his stressing the so- called intuitive necessity of the geometric axioms.

2. […]

In number 5114 of volume 214 of this journal, Mr. Anderson has raised several objections to my reply to Mr. Wulf. I have only now become aware of them, so my response will seem belated. All the same, I do not want to forgo this response because it is important to clarify the basis of this incessant misunderstanding of the theory of relativity.

Anderson admits that the theory of relativity provides a contradiction-free explanation for the “relativistic” perspective of the carousel at rest and the stars rotating with large angular velocity. He believes, however, that the theory becomes flawed when the direction of motion reverses; he contends that it is a coincidence in the relativistic perspective that all stars reverse their direction of motion at the same time. Let me begin by saying that obviously the changes in the directions of motion are contained in the differential equations of the gμν-fields. Further, it is not a matter of chance that the motions of the stars define the celestial axis as a privileged straight line, but rather it is well grounded in the distribution of stellar motions; the same conditions that, from the non-relativistic viewpoint, place the Earth at rest, also determine the distribution of the stars from the relativistic perspective after an appropriate transformation. Hence, this can in no way be called a coincidence.

Since the dispute over the theory of relativity has begun to die down over the last several years and the new theory has been even more successfully worked through, the most recent attacks upon it have come from the flank from which it was the least expected. They are not attacks on the philosophical motivation, and thereby not the well-known reproaches that the theory is “inconceivable” or “incompatible with common senses”; rather, we are now confronted with a physical experiment that stands in explicit contradiction to an assertion of the theory of relativity. This experiment was conducted by the American D. C. Miller at Mount Wilson and was published in the Proceedings of the National Academy, Washington (11, 382, 1925).

It concerns the so-called Michelson experiment, one of the most foundational pillars upon which the theory of relativity is constructed. This experiment traces back to the ideas of Maxwell, but it was Michelson, a scholar famous for his precision in optical measurement, who first carried it out. Michelson had already begun his investigation in the seventies in Berlin as an assistant to Helmholtz, and carried it out in the eighties in America. We can describe the experiment in schematic form in the following way (Fig. 13.1): two rigid arms are placed at right angles with mirrors S1 and S2 fixed perpendicularly to the arms at their end points.