CHAPTER OVERVIEW

The Lagrangian and Hamiltonian formalisms of particle dynamics can be generalized and extended to describe continuous systems such as a vibrating rod or a fluid. In such systems each point x, influenced by both external and internal forces, moves independently. In the rod, points that start out very close together always remain close together, whereas in the fluid they may end up far apart. The displacement and velocity of the points of the system are described by functions ψ(x, t) called fields.

In this chapter we describe the classical (nonquantum) theory of fields. Particle dynamics will be transformed to field theory by allowing the number of particles to increase without bound while their masses and the distance between them go to zero in such a way that a meaningful limit exists. This is called passing to the continuum limit.

LAGRANGIAN FORMULATION OF CONTINUUM DYNAMICS

PASSING TO THE CONTINUUM LIMIT

THE SINE–GORDON EQUATION

In this section we present an example to show how to pass to the continuum limit. Consider the system illustrated in Fig. 9.1, consisting of many simple plane pendula of length l and mass m, all suspended from a horizontal rod with a screw thread cut into it. The planes of the pendula are perpendicular to the rod (only two of the planes are drawn in the figure), and the pendula are attached to (massless) nuts that move along the rod as the pendula swing in their planes [for more details, see Scott (1969)].

This review concentrates on nucleosynthesis processes in general and their applications to massive stars and supernovae. A brief initial introduction is given to the physics in astrophysical plasmas which governs composition changes. We present the basic equations for thermonuclear reaction rates and nuclear reaction networks. The required nuclear physics input for reaction rates is discussed, i.e. cross sections for nuclear reactions, photodisintegrations, electron and positron captures, neutrino captures, inelastic neutrino scattering, and beta–decay half–lives. We examine especially the present state of uncertainties in predicting thermonuclear reaction rates, while the status of experiments is discussed by others in this volume (see M. Wiescher). It follows a brief review of hydrostatic burning stages in stellar evolution before discussing the fate of massive stars, i.e. the nucleosynthesis in type II supernova explosions (SNe II). Except for SNe la, which are explained by exploding white dwarfs in binary stellar systems (which will not be discussed here), all other supernova types seem to be linked to the gravitational collapse of massive stars (M>8M⊙) at the end of their hydrostatic evolution. SN1987A, the first type II supernova for which the progenitor star was known, is used as an example for nucleosynthesis calculations. Finally, we discuss the production of heavy elements in the r–process up to Th and U and its possible connection to supernovae.

CHAPTER OVERVIEW

Most dynamical systems cannot be integrated in closed form: the equations of motion do not lend themselves to explicit analytic solution. There exist strategies, however, for approximating solutions analytically and for obtaining qualitative information about even extremely complex and difficult dynamical systems. Part of this chapter is devoted to such strategies for Hamiltonian dynamics.

The first two sections describe Hamilton–Jacobi (HJ) theory and introduce action–angle (AA) variables on T*ℚ. HJ theory presents a new and powerful way for integrating Hamilton's canonical equations, and motion on T*ℚ is particularly elegant and easy to visualize when it can be viewed in terms of AA variables. Both schemes are important in their own right, but they also set the stage for Section 6.3.

THE HAMILTON–JACOBI METHOD

It was mentioned at the end of Worked Example 5.5 that there would be some advantage to finding local canonical coordinates on T*ℚ in which the new Hamiltonian function vanishes identically, or more generally is identically equal to a constant. In this section we show how to do this by obtaining a Type 1 generating function for a CT from the initial local coordinates on T*ℚ to new ones, all of which are constants of the motion. The desired generating function is the solution of a nonlinear partial differential equation known as the Hamilton–Jacobi (or HJ) equation. Unfortunately, the HJ equation is in general not easy to solve.

CHAPTER OVERVIEW

This chapter is devoted almost entirely to nonlinear dynamical systems, whose equations of motion involve nonlinear functions of positions and velocities. We concentrate on some general topics, like stability of solutions, behavior near fixed points, and the extent to which perturbative methods converge. Because nonlinear systems involve complicated calculations, the chapter stresses mathematical detail. It also presents results of some numerical calculations. The importance of numerical methods can not be overemphasized: because nonlinear systems are inherently more complicated than linear ones, it is often impossible to handle them by purely analytic methods. Some results depend also on topics from number theory, and these are discussed in an appendix at the end of the chapter.

We have already dealt with nonlinear systems perturbatively (e.g., the quartic oscillator of Section 6.3.1), but now we will go into more detail. It will be shown, among other things, that in the nonperturbative regime nonlinear systems often exhibit the kind of complicated behavior that is called chaos.

The first four sections of the chapter do not use the Lagrangian or Hamiltonian description to deal with dynamical systems. In them we discuss various kinds of systems, largely oscillators, which are often dissipative and/or driven. We go into detail concerning some matters that have been touched on earlier (e.g., stability, the Poincaré map), introducing ideas and terminology that are important in understanding the behavior of nonlinear systems of all kinds. In particular, considerable space is devoted to discrete maps.

Among the first courses taken by graduate students in physics in North America is Classical Mechanics. This book is a contemporary text for such a course, containing material traditionally found in the classical textbooks written through the early 1970s as well as recent developments that have transformed classical mechanics to a subject of significant contemporary research. It is an attempt to merge the traditional and the modern in one coherent presentation.

When we started writing the book we planned merely to update the classical book by Saletan and Cromer (1971) (SC) by adding more modern topics, mostly by emphasizing differential geometric and nonlinear dynamical methods. But that book was written when the frontier was largely quantum field theory, and the frontier has changed and is now moving in many different directions. Moreover, classical mechanics occupies a different position in contemporary physics than it did when SC was written. Thus this book is not merely an update of SC. Every page has been written anew and the book now includes many new topics that were not even in existence when SC was written. (Nevertheless, traces of SC remain and are evident in the frequent references to it.)

From the late seventeenth century well into the nineteenth, classical mechanics was one of the main driving forces in the development of physics, interacting strongly with developments in mathematics, both by borrowing and lending. The topics developed by its main protagonists, Newton, Lagrange, Euler, Hamilton, and Jacobi among others, form the basis of the traditional material.

This chapter is a review of the background and status of several current problems of interest concerning cosmic rays of very high energy and related signals of photons and neutrinos.

Introduction

The steeply falling spectrum of cosmic rays extends over many orders of magnitude with only three notable features:

1. (a) The flattened portion below 10 GeV that varies in inverse correlation with solar activity,

2. (b) The “knee” of the spectrum between 1015 and 1016 eV, and

3. (c) the “ankle” around 1019 eV.

4. For my discussion here I will divide the spectrum into three energy regions that are related to the two high–energy features, the knee and the ankle: I: E < 1014 eV, II: 1014 < E < 1018 eV and III: > 1018 eV.

In Region I (VHE) there are detailed measurements of primary cosmic rays made from detectors carried in balloons and on spacecraft. These observations, and related theoretical work on space plasma physics, form the basis of what might be called the standard model of origin of cosmic rays. Cosmic rays are accelerated by the first order Fermi mechanism at strong shocks driven by supernova remnants (SNR) in the disk of the galaxy. The ionized, accelerated nuclei then diffuse in the turbulent, magnetized plasma of the interstellar medium, eventually escaping into intergalactic space at a rate that depends on their energy.

The structure of neutron stars is discussed with a view to explore (1) the extent to which stringent constraints may be placed on the equation of state of dense matter by a comparison of calculations with the available data on some basic neutron star properties; and (2) some astrophysical consequences of the possible presence of strangeness, in the form of baryons, notably the Λ and Σ−, or as a Bose condensate, such as a K− condensate, or in the form of strange quarks.

Introduction

Almost every physical aspect of a neutron star tends to the extreme when compared to similar traits of other commonly observed objects in the universe. Stable matter containing A ∼ 1057 baryons and with a mass in the range of (1 − 2) M⊙ {M⊙ ≅ 2 × 1033 g) confined to a sphere of radius R ∼ 10 km (recall that R⊙ = 6.96 × 105 km) represents one of the densest forms of matter in the observable universe. Depending on the equation of state (EOS) of matter at the core of a neutron star, the central density could reach as high as (5 − 10)p0, where p0 ≅ 2.65 × 1014 g cm−3 (corresponding to a number density of n0 ≅ 0.16 fm−3) is the central mass density of heavy laboratory nuclei (compare this to P⊙= 1.4 g cm−3).

In its treatment of dynamical systems, this book has emphasized several themes, among them nonlinearity, Hamiltonian dynamics, symplectic geometry, and field equations. The last section on the Hamiltonian treatment of nonlinear field equations unites these themes in bringing the book to an end. To arrive at this point we have traveled along the roads laid out by the Newtonian, Lagrangian, Hamiltonian, and Hamilton–Jacobi formalisms, eventually replacing the traditional view by a more modern one. On the way we have roamed through many byways, old and new, achieving a broad understanding of the terrain.

The end of the book, however, is not the end of the journey. The road goes on: there is much more terrain to cover. In particular, statistical physics and quantum mechanics, for both of which the Hamiltonian formalism is crucial, lie close to the road we have taken. These fields contain still unsolved problems bordering on the matters we have discussed, in particular questions about ergodic theory in statistical physics and about chaos in quantum mechanics. As usual in physics, the progress being made in such fields builds on previous work. In this case it depends on subject matter of the kind treated in this book.

Although they are deterministic, classical dynamical systems are generically so sensitive to initial conditions that their motion is all but unpredictable. Even inherently, therefore, classical mechanics still offers a rich potential for future discoveries and applications.

CHAPTER OVERVIEW

This chapter contains material that grows out of and sometimes away from Lagrangian mechanics. Some of it makes use of the Lagrangian formulation (e.g., scattering by a magnetic dipole, linear oscillations), and some does not (e.g., scattering by a central potential, chaotic scattering). Much of this material is intended to prepare the reader for topics to be discussed later: chaotic scattering off hard disks is a particularly understandable example of chaos, and linear oscillators are the starting point for the discussion of nonlinear ones, of perturbation theory, and field theory.

SCATTERING

SCATTERING BY CENTRAL FORCES

So far we have mostly concentrated on bounded orbits of central-force systems. Yet unbounded orbits are very common for both attractive and repulsive central forces. Indeed, for repulsive ones all orbits are unbounded. For attractive ones, the outer turning point of bounded orbits – the maximum distance of the particle from the attracting center – increases with increasing energy. If the force is weak enough at large distances, the particle may escape from the force center even at finite energies: the orbit may break open and the outer turning point may move off to infinity. Then the orbit becomes unbounded, as in the hyperbolic orbits in the Coulomb potential. The result is what is called scattering.

GENERAL CONSIDERATIONS

Consider a central-force dynamical system that possesses unbounded orbits. Let the initial conditions be that a particle is approaching the force center from far away, so far away that it may be thought of as coming from infinity, which means that it is moving on one of the unbounded orbits.

This paper presents a discussion of the characteristic observables of stellar explosions and compares the observed signatures such as light curve and abundance distribution with the respective values predicted in nucleosynthesis model calculations. Both the predicted energy generation as well as the abundance distribution in the ejecta depends critically on the precise knowledge of the reaction rates and decay processes involved in the nucleosynthesis reaction sequences. The important reactions and their influence on the production of the observed abundances will be discussed. The nucleosynthesis scenarios presented here are all based on explosive events at high temperature and density conditions. Many of the nuclear reactions involve unstable isotopes and are not well understood yet. To reduce the experimental uncertainties several radioactive beam experiments will be discussed which will help to come to a better understanding of the correlated nucleosynthesis.

Introduction

Historically, the field of nuclear astrophysics has been concerned with the interpretation of the observed elemental and isotopic abundance distribution (Anders & Grevesse 1989) and with the formulation and description of the originating nucleosynthesis processes (Burbidge et al. 1957; Wagoner 1973; Fowler 1984). Each of these nucleosynthesis processes can be characterized by a specific signature in luminosity and/or in the resulting abundance distribution.

The Mexican School on Nuclear Astrophysics was held in the Hotel Castillo Santa Cecilia, Guanajuato, México, from August 13 to August 20, 1997. The goal of the school was to gather together researchers and graduate students working on related problems in astrophysics – to present areas of current research, to discuss some important open problems, and to establish and strengthen links between researchers. The school consisted of eight courses and material presented in these forms the basis of this book.

Non–stop interaction between the participants, through both formal and informal discussions, gave the school a relaxed and productive atmosphere. It provided the opportunity for researchers from a wide range of backgrounds to share their interests in and different perspectives of the latest developments in astrophysics.

The productivity of the meeting reflected the strong interest of the Mexican and Latin American scientific communities in the subjects covered, Indeed, a second school is planned for 1999.

Professor David Schramm very sadly died not long after the conference, in December 1997. His lectures at the School were fascinating. He will be sorely missed by us and the rest of the astrophysics community.