The equations of motion in classical physics differ considerably depending upon the subject they describe: a particle, an electromagnetic field, or a fluid. However our natural yearning for unification in the description of different phenomena has long since led to the development of universal formalisms. Among these the Lagrangian and Hamiltonian formalisms are the most advanced. This can be explained by the nature of the phenomena discussed. The popularity of each method varied at different stages in the development of physics. Throughout the whole period of advancement of relativistically invariant theories, preference was chiefly given to Lagrangian formalism (this was most conspicuous in field theory and the theory of a continuous medium). To a large extent, it was not before the generalization of the concepts of Hamiltonian formalism and introduction of Poisson's brackets that the Hamiltonian method of analysis was able to compete with the Lagrangian one.

The formation of new ideas and possibilities triggered recently by the discovery of the phenomenon of dynamic stochasticity (or simply, chaos) have brought the methods of Hamiltonian dynamics to the fore. Liouville's theorems on the conservation of phase volume and on the integrability of systems with a complete set of integrals of motion have determined both the formulation of many problems of dynamics and the methods of their study. The Hamiltonian method turned out to be of extreme importance for the theory of stability, which was advanced in this direction by Poincaré. Numerous subsequent studies have shown that Hamiltonian systems (i.e., systems which can be described by Hamiltonian equations of motion) demonstrate fundamental physical differences from other (non-Hamiltonian) systems. This chapter provides the most necessary information on Hamiltonian systems.

So far, we have been discussing various kinds of pattern with regular or almost regular symmetry. They emerged either in phase space of dynamic systems or in coordinate space of hydrodynamic flows. Common to all these cases was the method of obtaining or revealing patterns. Such patterns emerged not as the result of some artificial formal algorithm but as an expression of natural laws. In ancient times, however, people did not possess the level of knowledge available to us today. Perhaps it was the attempt to penetrate into the laws of creation of regular patterns, that gave rise to the art of ornament. Or perhaps this form of human activity had nothing to do with what was observed in nature. In either event, it would be interesting to make a number of comparisons between ancient ornaments and the pictures drawn by the trajectory of a real particle under certain conditions.

Two-dimensional tilings in art

Byzantine mosaic is one of the oldest examples of symmetrical periodic tilings of a plane (Fig. 10.1.1). Although the periodicity condition might have arisen as an independent problem, practical aims of architectural design required exactly this kind of ornament. Tiles of one shape (or of several different shapes) were to form the elementary components of a tiling. The element of an ornament was to be reproduced as many times as need, so that eventually any chosen portion of the plane could be paved.

The ornamental technique reached its peak of development in Muslim art. Elementary cells of an ornament became far more complex (Figs. 10.1.2 to 10.1.4).

We have shown that the phase portrait of a dynamic system with 1½ degrees of freedom can have a certain not-too-complex periodical or almost periodical structure. Such properties of the system's phase plane require quite definite conditions, which imply the existence of a stochasticity region (a stochastic web). It was this web that caused a partitioning of the phase plane either with a periodic symmetry (or simply symmetry) or with a quasi-periodic symmetry (or quasi-symmetry). This led us to a new concept of symmetries produced by dynamic systems in phase space. To what extent are symmetries of partitioning of phase space universal and are similar symmetries found in other natural objects?

We have already mentioned quasi-crystals as an example of quasi-symmetrical patterns. In this chapter we are going to show that hydrodynamic flows can also possess symmetry and quasi-symmetry similar to those displayed by the dynamic systems discussed earlier.

The formation of symmetry patterns in liquids has been known for a long time. An example of one of the simplest patterns (from the geometrical point of view) is Von Karman's vortex street. It consists of two rows of evenly spaced point vortexes arranged as on a chess board. The model of the vortex street was proposed by Von Karman to explain periodic traces behind a streamline cylinder for Reynolds numbers 30 < Re < 300. Von Karman's street is an example of a one-dimensional periodic chain, and this structural property is widely used in the analysis of its characteristics.

Among two-dimensional periodic patterns, the most famous is Benard's convective cell. This cell is formed as a result of thermal convection (the so-called Rayleigh–Benard convection).

Situations in which the system's integrability is immediately evident are very rare. Far more often, one has to do with a problem where a small perturbation is added to an otherwise integrable system. A large variety of methods and developed techniques exist to help analyse the effect of a perturbation on a system. Each is, in one way or another, related to the problem of a system's stability. Here one should remember that both the formulation of the stability problem and the very concept of stability are highly conditional. Any particular question concerning the system's properties determines not only the method of solution, but a specific definition of stability as well. For instance, the small perturbations may result in a weak response of the system's parameters either during a certain finite time interval or during an infinite time. In the latter case, the phase space topology of the system may either remain unchanged or may change considerably, etc.

Recently, studies on the stability of dynamic systems have been enriched by new methods and concepts. This is due to the discovery of a new phenomenon – dynamic chaos (or, simply, chaos). It turns out that while the equations may show no trace of random sources, the trajectories of a system can be the result of random time-dependent processes. The property of chaos characterizes the system as non-integrable. It is inherent only in nonlinear systems.

In this chapter we shall provide the most essential information on stability and chaos in dynamic systems. (Note 2.1)

Nonlinear resonance

Nonlinear resonance is an essential property of dynamic systems.

In the previous chapter we showed that in two-dimensional hydrodynamics there can exist flows with patterns displaying symmetry and quasi-symmetry. However, the picture drawn in the previous chapter is poor compared to the one we are now going to present. Three-dimensional dynamics introduces us to a qualitatively new phenomenon – the existence of stream lines chaotically arranged in space – which is sometimes called the Lagrangian turbulence. Various forms of this phenomenon have interesting practical applications and have played an important role in our understanding of the onset of turbulence, as well. In this chapter we are going to establish the relation between the structural properties of steady-state three-dimensional flows and the chaos of stream lines in these flows. This relation will sufficiently improve our understanding of those domains of physics where a stochastic web is mentioned. At the same time, we shall notice universality in the manifestations of quasi-symmetry in physical objects as different as the phase portrait of a dynamic system in phase space and the geometrical pattern of a steady-state flow of a liquid in coordinate space.

Stream lines in space

Stream lines of two-dimensional flows have a very simple structure and coincide with lines of the level of the stream function ψ(x, y). The behaviour of stream lines in steady-state three-dimensional flows can be completely different, since three-dimensional dynamics differs drastically from two-dimensional dynamics.

The physical phenomena taking place in condensed matter are far more diverse than phenomena produced by the motion of discrete particles. As the number of degrees of freedom increases, the need arises for a qualitatively new analysis of the dynamics of matter. One of the important manifestations of the difference between these two cases is the formation of regular patterns in matter. Their examples are many and varied: crystals and vortex streets, convective cells and high clouds, fracture patterns and soap foam, the arrangement of seeds in a sunflower head and Jupiter's Great Red Spot, snowflake patterns and the ‘packing’ of live cells. The list could be continued indefinitely. It is only natural that we should yearn for universality in understanding all these phenomena. Following the recent discovery of quasi-crystals, we understand much more about the possible types of patterns and symmetries. Speaking of the symmetry of some particular object, we imply that it possesses a certain invariant property. Although in the analysis of spatial patterns and their symmetries, the use of geometrical methods is quite natural, dynamic methods might come in useful as well.

As demonstrated above, the dynamics of particles and fields can also exhibit certain symmetries. In this chapter we shall learn in which way symmetry in dynamic problems can help achieve and analyse symmetry in spatial patterns. For various reasons, the existence of this relation proves very fruitful and enables deeper penetration into the nature of things. Recall what efforts and ingenuity it took Kepler to convince his readers of the existence of certain formative forces and factors. Spatial patterns and laws of their growth are determined by interaction (Note 7.1).

How does the onset of chaos in Hamiltonian systems occur? This is one of the key questions in the modern theory of dynamic systems. However narrowly specialist this question may seem, the answer has a bearing on almost every branch of physics, including the quantum theory.

Chaos emerges as a result of specific local instability with respect to arbitrarily small perturbations of the system's orbits. It manifests itself in certain regions of phase space and within a certain range of the system's parameters. But the most remarkable feature of chaos in the fact that it is irremovable in fairly general physical situations. What is meant is the following. Under fairly typical conditions in phase space and in the space of values of parameters there always exist such regions in which the dynamics of the system is stochastic. These regions may be arbitrarily small, nevertheless, for a certain structure of the dynamic system given by its Hamiltonian, they are irremovable at any finite values of parameters. An illuminating example of this situation is Arnold's diffusion – a universal, unlimited transport of particles along the channels of a stochastic web in systems with the number of degrees of freedom exceeding two.

As we transfer from systems totally free of stochastic dynamics to systems with chaos, we encounter small regions which are seeds of chaos. In Hamiltonian systems these are stochastic layers and stochastic webs which, being the manifestation of weak chaos in these systems, at the same time perform a certain partitioning of phase space.

The picture of the onset of chaos, as we see it today, is so extremely complex that one is advised to assume a step-by-step approach in its comprehension, temporarily omitting certain questions from consideration. In the preceding chapter, nothing was said of the border between the stochastic layer and the region of invariant curves. For that reason, in calculating the width of the stochastic layer, we adopted the approximate inequality K ≳ 1 as the border of chaos. Of course, the question of the conditions of the onset of chaos emerged as soon as the first studies on the analysis of real physical systems appeared. The word ‘real’ here means ‘typical’ for many physical problems, since there are strict mathematical criteria of chaos which can be illustrated by not-too-abstract models (Note 4.1). The main feature shared by these ‘chosen’ cases is the absence in phase space of sandwich-like or hole-riddled structures in which regions of chaos alternate with regions of stability. Examples of the first and second type are presented in Figs. 3.4.1 and 4.0.1, respectively: the ‘stochastic sea’ (the shaded area) is made up of points of the mapping belonging to a single trajectory, while the light regions are islands that cannot be reached by a trajectory from the sea region, and vice versa.

The coexistence of stability regions and regions of chaos in phase space presents severe problems in the study of dynamical chaos (Note 4.2). This difficulty comes up, for example, in the structure of the stochastic sea.

Introduction

From the results of the previous chapters we have seen that, under suitable assumptions (the compressibility hypothesis) a relativistic compressive nonlinear acoustic or magnetoacoustic pulse steepens and degenerates into a shock wave. Therefore, shocks are common occurrences in nonlinear wave motion and this chapter is devoted to laying out the basic theory of relativistic shock waves. Relativistic shocks are a very important feature in several models of phenomena occurring in astrophysics, plasma physics, and nuclear physics and in the following we shall briefly touch upon some examples.

Supernovas represent one of the most fierce phenomena occurring in the universe. A star suddenly increases its luminosity by many orders of magnitude such that at its maximum its light can outshine the total light from its parent galaxy. This phenomenon is suggestive of an explosion taking place in the star. Several mechanisms have been proposed in order to explain the source of energy driving the explosion (Carbon detonation, neutrino energy deposition, gravitational collapse and bounce, etc.).

In the case of massive stars, in the range between 8 and 100 solar masses, which are thought to be progenitors of type II supernovas, one of the most viable mechanisms for producing an explosion is gravitational collapse and bounce (Van Riper, 1979). At the end of stellar evolution the star will develop a core composed mainly of nuclei near the iron peak and free electrons, with a mass close to the Chandrasekhar limit of about 1.4 solar masses.