Saturday, September 11, 2010

Book Review: Mechanics, by Landau and Lifshitz

It is my intent to review every book in my store, below, so I will start with the first book in the store. You can purchase this book at my book store below.

This is the first volume of the classic Course of Theoretical Physics due to Lev Landau and the Landau Institiute of Theoretical Physics in Russia. These ten volumes were the subject of what Landau belived to be the required preparation in physics for any theoretical physicist.

This volume begins with a touching introduction to Lev Landau and his philosophy. It also lays out the prerequisites for this volume. To quote: "...ability to solve any indefinite integral that can be expressed in terms of elementary functions and to solve any ordinary differential equation of the standard type, knowledge of vector analysis and tensor algebra as well as the principles of the theory of functions of a complex variable (theory of residues, Laplace method). If you need this ackground I recommend Hassani, Mathematical Methods for Students of Physics and Related Fields, Second Edition.

The first chapter describes Lagrange's equations of motion from the starting point of generalized coordinates. This lays the ground work for a very nice presentation of the least action principle. Then there is a nice section of the Galilean transformations; it is important to realize that there is no discussion of special relativity in the book. The book then describes the Lagrangian of a free particle in Cartesian, spherical, and cylindrical coordinates. The final section of the first chapter not only discusses the Lagrangian of a system of particles, but nicely links Lagrange's equations to Newtonian theory.

The second chapter presents one of the most important aspects of theoretical physics, the conservation laws. This begins with a unified treatment of integrals of the motion, the conservation of energy, how conservation of energy implies the homogeneity of time, and conservative systems. This leads into the conservation of momentum, how the conservation of momentum implies the homogeneity of space, generalized momenta, and generalized forces. There is a brief discussion of systems of particles, the center of mass, and internal energy. This leads to a section on the conservation of angular momentum, how this implies the isotropy of space, and the idea of the central field. The final section deals with a couple of different topics such as the behavior of the equations of motion under transformations and the virial theorem.

The third chapter is a very practical one about applying the equations of motion to specific situations. This begins with one dimensional motion and a discussion of energy diagrams and oscillations. This leads into a discussion of how to interpret energy diagrams and derive the potential from a specific period of oscillation. The next topic covered is the beginning of the two-body problem in the form of the reduced mass of a system. The two-body problem is then reduced to a one-body problem in a central field in a very clear presentation. This leads to a nice discussion of the Kepler problem for bound and unbounded orbits.

Chapter four is an important discussion of the classical theory of particle scattering. This begins by the idea that particles can break up, this also introduces the lab and center-of-mass frames of reference. Then the authors describe elastic collisions in quite intuitive way. Then the notions of impact parameters and scattering-cross sections are covered. As an example of scattering due to fields, there is a section on Rutherford scattering. The chapter ends with small-angle scattering.

Chapter Five is on the theory of oscillations. Naturally, this begins with free oscillations in one dimension. The book then turns its attention to forced oscillations and resonance. Then the authors treat the idea of oscillations in more than one degree of freedom, including eigenfrequences and normal modes. This is then applied to the classical theory of molecules. Then damped oscillators are covered along with a discussion of dissipative functions in Lagrangian dynamics. The next section introduces dissipation and damped and driven oscillations. At this point the book leaves the realm of simple oscillating systems and goes into parametric oscillations and the Mathieu equation. There is a discussion of nonlinear oscillators, laying the ground-work for the study of chaos, this also includes a section of the resonance of nonlinear oscillators—introducing the Duffing oscillator without calling it that. Finally, there is an interesting section on the motion of a particle in a rapidly oscillating field.

Chapter six is a thorough treatment of rigid body dynamics. This begins with a discussion of the kinematics of rigid bodies. Then the authors introduce the inertia tensor. Indeed, this is one of the few sections in the book with more than a few problems (there are nine here). The section on angular momentum uses the axes of inertia to define angular momentum. Then the equations of motion for a rigid body are derived. This leads to the notion of Euler angles, which in turn leads to the Euler equations. These notions are then applied to the motion of a top. Then they shift gears a little and discuss rigid bodies in contact. Finally, and I think this might have done better at the beginning of the chapter, is a section on motion in noninertial frames, specifically rotating frames. This chapter seems very conventional, though it must be remembered that the first version of it came out in 1960, so a lot of other books are based on material found in here.

The final chapter is an overview of Hamiltonian theory. This begins, reasonably enough, with Hamilton's canonical equations of motion and relates the Hamiltonian and the Lagrangian in the normal way. The Routhian has its own section. Then they address the extremely important topic of the Poisson brackets. The next section establishes the idea of a functional without using those words. Then the authors discuss the principle of Maupertuis. Then they discuss canonical transformations and link them to Poisson brackets, and discuss conjugate variables, laying some of the groundwork for quantum mechanics. The authors next introduct the idea of phase space through a discussion of Liouville's theorem. The next section discusses the Hamilton-Jacobi equation and the idea of general and complete integrals. This is followed by a treatment of the method of separation of variables. Then the authors have an excellent and clear discussion of adiabatic invariants, a subject with applications in plasma physics. Then they discuss action-angle variables. Then the authors turn to the validity of the adiabatic invariant. The chapter, and the book, ends with a section on conditionally periodic motion.

The book, with all problems included, is just 167 pages long and comprises a total of 52 sections. This is something you could easily cover in two months of dedicated study. I think the book could use an update, but it's a classic as it is and its choice of topics are pretty good. I like it a lot, even if it doesn't really cover chaos in any meaningful way.

1 comment: