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Title: Classical Mechanics by Richard A. Matzner, Lawrence C. Shepley ISBN: 0-13-137076-6 Publisher: Prentice Hall Pub. Date: 01 April, 1991 Format: Hardcover List Price(USD): $88.00

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Average Customer Rating: 3 (1 review)

Rating: 3
Summary: Close to the edge of modern mechanics.
Comment: Classical mechanics has certainly changed in form since the days of Newton, due in part to the Lagrangian and Hamiltonian formulations, and to the rise of the theory of relativity. Student pursuing graduate study in mechanics will be exposed to differential geometry as well as other more abstract mathematics. Formulating mechanics using these mathematical tools has the advantage that it remains loyal to the spirit of the theory of relativity, especially the general theory, which makes heavy use of differential geometry. It has the disadvantage of not being amenable to numerical computation, and sometimes masks the underlying intuition.

This short book proposes to be a "modern" introduction to classical mechanics, and it succeeds in its goal to a large degree. However, there is no discussion at all of chaos, in spite of its importance and modernity. Chaotic mechanical systems, although only discovered recently in comparison to the long history of mechanics, can still be thought of in the context of classical (Newtonian) mechanics, even though they are more easily formulated in the Hamiltonian formalism.

Chapter 1 of the book is an introduction to the differential geometry to be used in the book, but not from a rigorous mathematical standpoint. The treatment is done in the context of differentiable manifolds, with tangent spaces and bundles, gradients and 1-forms, and tensors all being defined, albeit somewhat hurriedly. Newtonian kinematics is formulated in the tangent bundle, called the configuration bundle. The kinetic energy is shown to be a tensor on this bundle. The theory of constrained mechanical systems is discussed, but only in the context of holonomic constraints.

In chapter 2, the authors show how to formulate Newtonian physics in the context of Minkowski space. Due to the assertion that time is absolute in the Newtonian theory, the time slices in this space are set equal to each other. Thus the indefinite metric property of Minkowski space does not play a role in this formulation.

The Lagrangian formulation of mechanics is outlined in chapter 3, wherein the authors show that Newton's equations can be obtained via the Lagrangian function, which is a scalar function on the tangent bundle. Defining the action integral over the Lagrangian and finding its extrema with fixed endpoints in time gives the Euler-Lagrange equations of motion. A particular choice of the Lagrangian then yields the Newtonian equations of motion. This is certainly a different formulation than what Newton had in mind, but it does have the advantage of mathematical simplicity, and it makes the transition to quantum physics much more palatable in some cases.

The authors turn to the consideration of central force fields in chapter 4, in the context of the Lagrangian formulation. Using conservation of total linear momentum and total angular momentum, the 6-dimensional configuration space of the two-body problem is reduced to two dimensions. Kepler's second law results from the equations of motion, as expected. The Coulomb force and classical scattering theory is then studied. A brief but interesting overview of the (restricted) three-body problem is then given.

In the next two chapters, the authors concentrate on matters of a purely mathematical nature. One of them concerns the uniqueness of the Lagrangian, which they show it is not. One can add a total time derivative to it and multiply it by a constant and still obtain the same equations of motion as the first. They then give a general discussion of when two Lagrangians are equivalent. They also discuss rotations in Euclidean 3-space, and, curiously, introduce spinors in order to parametrize fully 3-dimensional rotations. This is unusual in a text on classical mechanics, and unnecessary to a large extent.

The discussion of rotations does set up the treatment in chapter 7 on rigid body dynamics, wherein the authors derive the equations of motion in the body frame. This leads to a discussion of the particle dynamics from the standpoint of rotating frames, which lead to the famous (non-Newtonian) contributions to the total force: the Coriolis and centrifugal forces.

Mechanics from a "symplectic" viewpoint, namely the Hamiltonian formalism, is outlined in Chapter 8. The equations of motion in this formalism are first-order and involve the Hamiltonian function, which is obtained from the Lagrangian via a Legendre transformation. Ubiquitous now in the study of dynamical systems, especially ones that exhibit chaotic behavior, the Hamiltonian formalism, especially in the context of symplectic geometry, is one that has grown in importance in purely mathematical questions. The Poisson bracket is introduced in the next chapter, and the authors make the connection with symplectic geometry via the canonical transformations in chapter 10. The discussion here, will assist readesr in understanding the famous canonical quantization, which they will encounter in later courses on quantum physics.

The Hamilton-Jacobi theory, which is a formulation of mechanics in terms of a first-order nonlinear partial differential equation, and which is also very important in modern formulations of mechanics, and in quantum physics, is discussed in chapter 11. In fact the authors point out briefly the connection with wave mechanics in the last section of the chapter. The utility of the Hamilton-Jacobi theory in solving mechanics problems is brought out in chapter 12. Some mechanical systems, the separable ones, allow simplification into action and angle variables, as the authors discuss in fair detail. Others however, defy such a separation, and require approximation techniques. Canonical perturbation theory, which is one of these techniques, is discussed in chapter 13. This is followed in chapter 14 by a discussion of coupled oscillations, and the transition to classical field theory is made.

The stage is now set for the reader to go to on to many fascinating topics in modern mechanics: ergodic and KAM theory, chaotic dynamical systems, and the interesting mathematics involved in these areas.