Classical dynamics is traditionally treated as an early stage in the development of physics, a stage that has long been superseded by more ambitious theories. Here, in this book, classical dynamics is treated as a subject on its own as well as a research frontier. Incorporating insights gained over the past several decades, the essential principles of classical dynamics are presented, while demonstrating that a number of key results originally considered only in the context of quantum theory and particle physics, have their foundations in classical dynamics.
Graduate students in physics and practicing physicists will welcome the present approach to classical dynamics that encompasses systems of particles, free and interacting fields, and coupled systems. Lie groups and Lie algebras are incorporated at a basic level and are used in describing space-time symmetry groups. There is an extensive discussion on constrained systems, Dirac brackets and their geometrical interpretation. The Lie-algebraic description of dynamical systems is discussed in detail, and Poisson brackets are developed as a realization of Lie brackets. Other topics include treatments of classical spin, elementary relativistic systems in the classical context, irreducible realizations of the Galileo and Poincaré groups, and hydrodynamics as a Galilean field theory. Students will also find that this approach that deals with problems of manifest covariance, the no-interaction theorem in Hamiltonian mechanics and the structure of action-at-a-distance theories provides all the essential preparatory groundwork for a passage to quantum field theory.
This reprinting of the original text published in 1974 is a testimony to the vitality of the contents that has remained relevant over nearly half a century.
Sample Chapter(s)
Chapter 1: Introduction: Newtonian Mechanics (108 KB)
Chapter 2: Generalized Coordinates and Lagrange's Equations (98 KB)
Chapter 3: The Hamilton and Weiss Variational Principles and the Hamilton Equations of Motion (124 KB)
https://doi.org/10.1142/9789814713887_fmatter
The following sections are included:
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Physics is the science of change and of measurement. The description, contemplation, analysis, and esthetics of change manifested as motion within a classical framework are classical dynamics…
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We now generalize the considerations of the preceding chapter to cover many particle systems, then write the equations of motion in a form valid for any choice of coordinates…
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One can visualize the development in time of a system described by the Lagrangian equations of motion in the following way (unless otherwise stated, we are considering the standard case). The number of generalized coordinates, equivalently the number of “degrees of freedom,” being k, the k-dimensional configuration space of the system is defined as a real space whose points are labeled by the k generalized coordinates q1, q2 ,⋯, qk. Each qs goes over a range determined by the physical meaning of that coordinate. Each point in configuration space corresponds to a possible configuration of the system at one instant of time. If the configuration of the system at the time t0 corresponds to the point Q0 in configuration space, and if we start the system going in a particular direction at Q0 by specifying the velocities at t0, then the Lagrangian equations of motion will completely determine the generalized coordinates qs(t) for all later times; that is, the configuration of the system at each instant t is determined. The representative point Qt in configuration space describes a certain class of trajectories in that space; the trajectory is fixed once one knows the starting point and direction and magnitude of velocity, that is, the initial configuration and velocities. Alternatively, we could specify the configurations Q1 and Q2 at an initial time t1 and a later time t2, and then the equations of motion (in general) pick out a unique trajectory leading from Q1 at t1 to Q2 at t2; this trajectory tells us with what velocities we should start the system at Q1 at time t1 to ensure its arrival at Q2 at time t2. In either case the velocities at each interior point on the trajectory are determined…
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We now consider the problem of proving the equivalence of the configuration space and phase space versions of the Weiss action principle. We do this indirectly by deriving the Hamiltonian equations of motion from the Lagrangian ones. Recall that the latter have been put into the following form:
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The study of the invariance properties of the Lagrangian and the Hamiltonian equations of motion is a very important aspect of the formal structure of classical dynamics. Because both systems of equations are equivalent to variational principles that can be stated to a large extent in a coordinate-independent way, we must expect these invariance properties to be present. In specific cases the invariance properties can be used to find particularly suitable coordinates in terms of which the equations of motion are more tractable. From a more general point of view it is desirable to display and understand the full transformation structure of the classical theory because it has many analogies with the structure of quantum mechanics. However, we are not able to explain this latter relationship in any detail here. We deal, first, with the Lagrangian description, and later with the Hamiltonian one…
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Consider those canonical transformations on a fixed 2k-dimensional phase space that are not explicitly time dependent. All such transformations taken together form a group with an infinite number of elements. That is to say, the following laws, which define a group, hold…
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The two equivalent ways of defining canonical transformations using either PB's or LB's. Eqs.(5.28) and (5.29), are both differential in character. These conditions involving the fundamental brackets have been shown to be completely equivalent to the invariance properties, under canonical transformations, of the PB's and LB's among arbitrary functions on phase space. There is another type of entity, namely, the collection of volume and surface elements of various dimensions in phase space, which are also invariant under canonical transformations. We turn now to a discussion of these expressions…
https://doi.org/10.1142/9789814713887_0008
We have explained in Chapter 4 how one can pass from the Lagrangian to the Hamiltonian form of mechanics, and vice versa, in the case of systems that belong to the standard case. The preceding three chapters have indicated the power of the machinery of canonical transformations to which one is led from the Hamiltonian formulation of the theory. The first systematic discussion of systems whose Lagrangians are not of the standard type was given by Dirac. It is interesting to examine these systems in a general way from two different points of view. First, the nature of the solutions of the Lagrangian equations of motion can be strikingly different from the standard case. Secondly, one would like to see how much of the structure of the Hamiltonian formulation can be retained, and even how the Lagrangian form can be recast in the Hamiltonian form. Indeed the primary aim of Dirac was to develop a standard technique for “Hamiltonising” a non-standard Lagrangian and to use the new Hamiltonian form for developing the quantum mechanics of such systems. Even within the framework of classical theory, however, it is satisfying to see how far one can carry the analysis for an arbitrary Lagrangian. We devote this chapter and part of the next one to an exposition of the Dirac theory.
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The purpose of this chapter is to analyze the properties of the Dirac bracket and of the transformations generated by it. As we have seen in the preceding chapter, to define the Dirac bracket we need to be given an even number of functions, ζm, on a 2k-dimensional phase space with canonical coordinates ωμ, with the property that the matrix of PB's ∥{ζm, ζm′}∥ is nonsingular. We assume, for simplicity, that this matrix is nonsingular in the entire phase space, and denote its inverse by…
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In the discussion so far we have considered only systems that have arbitrarily many, but a finite number of, degrees of freedom. Now we extend these considerations to cover some aspects of the treatment of dynamical systems with an infinite number of degrees of freedom. Many of the important and interesting dynamical systems of this type involve classical fields as the basic dynamical variables. Examples are the classical electromagnetic and gravitational fields. In Hamiltonian field theory the infinity of the degrees of freedom is a continuous infinity. All the formal developments of the preceding chapters, such as the Lagrangian and Hamiltonian descriptions, the PB and its generalizations, canonical transformation theory, and even the Dirac theory of constraints, can formally be carried over to the description of field systems. However, it is natural to expect that the very fact that we have an infinite number of degrees of freedom, countable or not, introduces new and novel mathematical (and hence physical!) features that were completely absent previously. Most of these new features have been encountered for the first time and studied from the quantum field theory point of view, but we would like to indicate here that they are equally relevant for classical mechanics. It turns out that the easiest way to exhibit these features, and the most transparent way, is to consider dynamical systems with a countable infinity of degrees of freedom. For this reason we adopt the following order of presentation of the material in this chapter. We first exhibit by means of simple examples some characteristic differences between systems with a finite and countably infinite number of degrees of freedom. Then we show formally that there is a correspondence between the cases of countably infinite and continuously infinite degrees of freedom. We then consider the treatment of fields by Lagrangian and Hamiltonian methods, and conclude by discussing the example of sound vibrations in an ideal gas…
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We have mentioned in previous chapters the important connection between dynamical variables in phase space and the one-parameter groups of canonical transformations that they generate. This connection is a feature of the Hamiltonian form of dynamics. For example (see the discussion in Chapter 6 on page 56) a constant of motion is a function A(q, p) on phase space that generates a group of canonical transformations under which the Hamiltonian of the system does not change. We have noted in Chapter 9 that for a constrained system, the presence of first-class constraints weakens the reciprocal relationship between dynamical variables and the corresponding groups of canonical transformations. It is worthwhile to discuss in some detail the transformationtheoretical significance of two very important dynamical variables that are defined for most systems of physical interest, these being the variables of linear and angular momentum. Such a discussion serves as a model for the later treatment of other important dynamical variables (introduced by the theories of Galilean and Poincaré relativity). In Newtonian particle mechanics, the primary significance of the linear and angular momenta is that they are constants of motion for certain kinds of forces between the particles, and their forms are derived from the form of the basic equations of motion. On the other hand, it turns out that in Hamiltonian dynamics it is more natural to define these variables by means of the transformations that they generate on the basic dynamical variables, the q's and p's, of the theory. One can also appreciate this relationship from the following point of view. In discussing Lagrangian and Hamiltonian field theory in the previous chapter, we noted that the continuous three-vector x that on the one hand enumerates the continuous infinity of coordinates ϕ(x, t) and canonically conjugate momenta π(x, t), is on the other hand to be identified (in most physically interesting examples) with points in three-dimensional Euclidean space. Thus we spoke of Lagrangian and Hamiltonian densities and of the energy density. This geometrical interpretation of the label x naturally leads to important consequences for a dynamical system of fields. Euclidean space and the relations of Euclidean geometry are unchanged under a certain set of operations on points in that space; these are the translations and rotations of three-dimensional space and all these operations taken together comprise the Euclidean group. If a physical system embedded in Euclidean space is described in the Hamiltonian form of dynamics, then the effects of translations and rotations on the system must be adequately described in the mathematical framework. We see in this chapter how the linear and angular momenta are intimately connected with the translational and rotational properties for systems permitting a Lagrangian and Hamiltonian description. We find that both for multiparticle and field systems we can in fact define these dynamical variables in terms of the effects of translations and rotations. But the meaning of the algebraic relationship between the different components of linear and angular momenta, as well as the complete group-theoretical relationship between them and the Euclidean group, becomes clear in later chapters…
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The primary purpose of this and the two succeeding chapters is to provide the reader with an adequate understanding of the properties of Lie groups, Lie algebras, and their realizations so that he can then follow the relevance and use of these things in classical Hamiltonian mechanics, some aspects of which are discussed in the last few chapters of this book. The treatment that we give here of these mathematical topics certainly cannot aspire to be a very complete one, because that would be out of place in a book devoted more to classical mechanics than to group theory. On the other hand, too brief a treatment of the mathematics would have made these chapters useless except for those who know the subjects already. In the choice of material covered, therefore, a compromise has to be made; we have included all that is needed for our later applications, in addition to those topics that make for cohesion and continuity of presentation. In this chapter, we discuss sets, groups, and topological groups and give the definition of a Lie group. Chapter 13 describes the structure of Lie groups in more detail and explains how one goes from a Lie group to its Lie algebra and vice versa. Chapter 14 describes the kinds of realizations of these abstract objects that are relevant in Hamiltonian mechanics, namely realizations of Lie algebras and Lie groups via PB's and canonical transformations, respectively…
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Consider a Lie group G in which coordinates have been introduced in some neighborhood N0 of the identity e. Thus, if a is an element of G in N0, it corresponds to a point in n-dimensional real Euclidean space with coordinates…
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Our major interest in Lie algebras and Lie groups is in their relation to classical mechanics. We are interested in utilizing the structural similarities by defining a correspondence of dynamics and Lie algebras, as well as in the expression of various kinds of symmetries in a given dynamical framework. We are thus led to introduce the concepts of realizations and representations…
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As examples of physically important Lie groups, we describe in this chapter the following groups:
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We want to examine now the formal structure of the Hamiltonian description of a dynamical system possessing symmetry under one of the groups G that we studied in the previous chapter. For simplicity, let us call any one of these groups a “relativity group.” We assume, of course, that the system being discussed does permit a Hamiltonian description; in practice this may be arrived at by starting from an action principle stated in terms of a Lagrangian to which we then apply the standard rules for going over to a Hamiltonian. When we say that a certain system “possesses symmetry under a group G,” we mean the following: there is a class of observers with associated space-time coordinate systems, related to one another by elements of G; there are appropriate rules for passing from the description of the system given by one observer to that given by another observer; with the use of these rules the equations of motion take the same form for all these observers. For the cases when G is the Galilei group or the Poincaré group, the subgroup T1 corresponding to time translations or shifts in the zero of time is already present in G. As we see in detail later, this has the consequence that the general rules that connect the descriptions given by two observers linked by a general element of G contain as a special case the equations of motion, or equations for development in time. But suppose we wish to discuss a system possessing rotational symmetry alone, that is, possessing symmetry under R(3), or, say, possessing symmetry under E(3). In such cases, the class of observers involved is such that they all use the same time coordinate, and only their spatial coordinate systems differ, being related by elements of R(3) or E(3). But we are still interested in the equations of motion in time for these cases; thus we agree to extend these groups formally by including the time translations T1 in directproduct fashion, and agree to write G for the groups R(3) ⊗ T1 or E(3) ⊗ T1 that we obtain in this way. That is, it is convenient to allow two observers not only to use different coordinates in space, but also different origins in time, and then the position of the equations of motion in the general analysis will be the same for all groups G…
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We study in this chapter some examples of classical mechanical realizations of the group R(3). We have already explained in Chapter 15 some of the important structural properties of this group and its Lie algebra. For applications to physics this is an important group, partly for its own sake and partly because it appears as a subgroup in the other space-time transformation groups For this reason, it is useful to include a discussion of the linear matrix representations of this group, although it would take us too far afield to prove all the interesting properties we mention. We also give a description of the connection between R(3) and the group SU(2) consisting of all complex unitary unimodular matrices in two dimensions. This is the group that describes the rotational behavior of “spinors” in contrast to the behaviors of vectors and tensors, which are adequately described by the group R(3). These matters, which are not exclusively related to classical mechanics, are taken up in the latter part of this chapter…
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Our study of the classical canonical realizations of the group E(3) is a short one. We begin with the basic Lie brackets for the Lie algebra of E(3). There are six basic elements, lj and dj, j = 1,2,3, the former generating rotations and the latter translations, and they obey…
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The Galilei group is the first of the two groups of space-time transformations to be discussed in this book. For brevity we refer to it as G. This group expresses the geometrical invariance properties of the equations of motion of a non-relativistic classical dynamical system when the system is isolated from external influences. This is so to the extent that these equations of motion are reasonable generalizations of the Newtonian equations of motion for multi particle systems experiencing interparticle forces. If the dynamics of the system can be conveyed via an action principle, so that a Lagrangian and therefore a Hamiltonian description is possible, then the system is described by a classical canonical realization of G…
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This chapter is devoted to the study of some aspects and examples of classical canonical realizations of the Poincaré group, or as it is also called, the inhomogeneous Lorentz group. For brevity, we write P for this group; for the homogeneous Lorentz group we use the abbreviation HLG or sometimes SO(3, 1) as we did in Chapter 15. In passing over from the Galilei group, which describes the invariance properties of the equations of motion of Newtonian mechanics, to P, what is retained is the idea that there is a privileged class of reference frames in nature, the inertial frames, in which the laws of particle mechanics are especially simple. In particular a particle “not acted on by any forces” follows a straight-line trajectory in any one of these frames. As in the case of G, two inertial frames in special relativity theory must be related to one another by a space-time translation, or by a spatial rotation, or by the fact that one of them moves with a uniform velocity relative to the other, or by a combination of all these. What distinguishes P from G is the way in which we relate the spacetime coordinates assigned to an event by two observers in two different inertial frames. In the case of G this relation is the one that is “reasonable” and “obvious” from the stand point of Newtonian mechanics (and in fact from common, everyday experience). In the case of P we have, on the other hand, the Lorentz transformation formulae discussed in detail in Chapter 15. These different formulae for the way the space-time coordinates of events change in going from one inertial frame to another give different structures to the group of all possible changes of inertial frame, leading to G in the one case and P in the other. The transformation formulae in the case of P contain the velocity of light in vacuum, c, in an essential manner; one can, in a physically meaningful way, think of G as the limiting form of P as c → ∞. (Of course, c is a fixed real number in any given system of units and never really goes to infinity; what is meant is that in specific situations if certain characteristic velocities are neglected in comparison with c, in a consistent way, then P is essentially replaced by G). For the most part, we assume that our system of units is such that c can be set equal to unity, so that the writing becomes easier…
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The structures of the Galilei and Poincaré groups have been abstracted from the transformation laws for the space and time coordinates of “events,” corresponding to the transition from one inertial observer to another. These transformation laws are essentially geometrical in character; thus we may regard the two groups G and P as describing two possible space-time geometries. (The geometrical flavor is probably more pronounced in the relativistic case, but both are capable of geometrical interpretation). Because of this, one has the feeling, especially in the relativistic case, that space and time coordinates “ought to be treated on the same footing”. The principle of relativity, be it Galilean or Poincaré, as applied to mechanics is usually stated as follows: “All equivalent observers deduce the same laws of physics”. Let us summarize very briefly the way in which this is implemented in our discussion in Chapter 16. In Hamiltonian mechanics, the laws of motion are stated in terms of PBs. On the other hand, working for definiteness in the Heisenberg picture, the specific “legal” definitions of dynamical variables in different frames yield different (but related) dynamical variables. Hence we look for relations or transformations between the dynamical variables given by legal definitions in one frame and those similarly given in another frame, which preserve PB relations. But these are simply canonical transformations. (This is certainly so for finite numbers of degrees of freedom, but only formally so otherwise.) Thus, given any two inertial observers, there are two sets of dynamical variables identified by the legal definitions in each frame, and there is a canonical transformation connecting the two sets. In effect, not only is the development in time given by the action of a canonical transformation, but so is any other conceivable change of inertial frame. (Here we think of a change in the zero of time as a particular but simple change of frame.) Because the set of all transformations from one observer to an equivalent one has the structure of G or P, our analysis shows that we must have a realization of the relativity group by means of canonical transformations. Thus we find precisely the condition under which a canonical system has relativistic invariance, that is, for which physical laws deduced by different observers are the same: namely, when we have a canonical realization of the appropriate relativity group…
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We have seen that the canonical formalism is incapable of describing a relativistic many-particle system including interactions, if manifest covariance is demanded. (In this chapter the only relativity group we are concerned with is the Poincaré group.) Such dynamical systems must be described by other means. For suitable types of interactions we may still be able to derive the basic equations of the system from an action principle, and in this chapter we study examples of precisely such systems. However, the action functional can no longer be expected to be the time integral of a suitable Lagrangian which is local in time, and hence the basic “equations of motion” also are not local in time. Interactions between particles in the Newtonian nonrelativistic (or better Galilean relativistic) domain are summarized in the notion of a potential that acts nonlocally (“at a distance”) in space at a given time. If this is to be generalized to the Poincaré-relativistic domain and the description is manifestly covariant, the nonlocality in space implicit in the notion of a potential automatically implies nonlocality in time. This comes in from the geometry of Lorentz transformations that makes simultaneity frame-dependent; in contrast to a nonrelativistic potential that can act at one instant of time in every frame, a “relativistic potential” acts at unequal times as well…
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In the preceding chapters we have treated various aspects of classical dynamics and have endeavoured to show that the structure of the theory is intimately related to an associative algebra with derivations. Transformations of dynamical systems are intimately associated not only with their time development but also with the “apparent” changes due to a change in the frame of reference. In many ways our approach to classical mechanics in this book has been conditioned by knowledge of developments in quantum mechanics. Without trying to explain in detail the quantum mechanical notions involved, we conclude this book by first discussing some of the analogies as well as the differences between classical mechanics and quantum mechanics, and then we mention the ways in which the latter has “opened our eyes” to aspects of the former…
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The following sections is included:
"This book is useful for someone who wants to learn classical dynamics, not with a view to solve specific problems of particles or rigid bodies, but to understand the basic mathematical structure which underlies it and its close relation to quantum theory? It is still the best short introduction to Dirac's constraint analysis. There are lessons that relativity and quantum theory have taught us, and looking at the classical dynamics with this perspective is hugely rewarding."
"The reprinting of the textbook after more than 40 years is a testimony to the vitality of classical dynamics with many accompanied topics that has remained relevant until now. The textbook will be useful for graduate students, university lecture in physics, and practicing physicists."