1R2. Analytical Mechanics. Foundations of Engineering Mechanics Series. - AI Lurie (Saint-Petersburg Polytechnic Inst, Russia). Springer-Verlag, Berlin. 2002. 864 pp. ISBN 3-540-42982-4. \$139.00.

Reviewed by W Schiehlen (Inst B of Mech, Univ of Stuttgart, Pfaffenwaldring 9, Stuttgart, 70550, Germany).

This book by AI Lurie is a most valuable contribution to the foundations of engineering mechanics. Even if the original Russian edition of the book was published in 1961, the English translation reviewed is still a topical textbook of theoretical mechanics. The book is based on the definition of a material body as a collection of particles resulting in a finite number of degrees of freedom. The continuum mechanics approach, which is more standard today even for rigid bodies, is only mentioned, but not considered in detail. Thus, the theory presented fits perfectly to multi-particle systems, rigid and elastic frameworks, and celestial mechanics. Rigid bodies and gyroscopes are also covered assuming that their inertia parameters are known.

Chapter 1 is devoted to the basic definitions of engineering systems of material particles characterized by constraints between the particles. Based on this concept, generalized coordinates, velocities and accelerations, redundant coordinates, quasi-velocities, and virtual displacements are introduced. As an example, a two-axle trolley is used.

The second chapter deals with rigid bodies kinematics. Direction cosines and Euler angles are presented with special emphasis on airplane and ship angles. From the vector of infinitesimal rotation, the angular velocity is derived. The matrix form for velocity and acceleration of a rigid body is given. Further, the relative motion with respect to generalized coordinates is analyzed. Applications include Cardan’s suspension and a body rolling on a fixed plane. Chapter 3 presents the theory of finite rotations of rigid bodies. Starting with Rodrigues formula, the corresponding Rodrigues and Hamilton parameters, the Euler-Chasles theorem, and the relation between finite rotations and Euler angles are introduced. A complex valued combination of Rodrigues-Hamilton parameters results in Cayley-Klein parameters. The angular velocity vector for finite rotations is also derived for Cayley-Klein’s parameters. The evaluation of the position of a self-excited rigid body serves as an example.

The fourth chapter specifies the basic dynamic quantities as kinetic energy, momentum and angular momentum, and the energy of accelerations required for Appel’s equations later on. All these quantities are based on the tensor of inertia discussed in detail. Examples deal with Cardan’s suspension, flywheels, rotor platforms, rolling spheres and rings, and the two-axle trolley. In Chapter 5, the mechanical work and the related potential energy are discussed resulting in generalized forces due to gravity and elasticity. With respect to space applications also the shape of the Earth is considered. The potential energy is evaluated for rod structures subject to bending, torsion, and compression. Dissipative forces due to friction and aerodynamic resistance are analyzed.

Chapter 6 presents Lagrange’s equations of the first kind. For ideal constraints without any dissipation, one gets the fundamental equations of dynamics also known as D’Alembert’s principle or Lagrange’s central equations, respectively. The examples deal with the equilibrium of systems of particles, in particular, rod structures and rigid bodies including friction. Then, in Chapter 7, Lagrange’s equations of the second kind are derived. The structure of these equations is discussed, and a geometrical interpretation of the motion of the system is given. Even if the reaction forces are removed, it is shown how generalized forces are retained and computed efficiently. Cyclic coordinates and the Routhian function are presented. Numerous examples are found throughout this chapter like rod systems, slider-crank mechanisms, and heavy tops.

In Chapter 8, special forms of differential equations for nonholonomic systems are considered. Based on quasi-velocities defined as linear forms of generalized velocities, the Euler-Lagrange differential equations are found, and Appell’s differential equations are presented. Applications include rolling spheres and rings as well as the two-axle trolley. Chapter 9 is devoted to the dynamics of relative motion. From the corresponding differential equations, the relative equilibrium of rotating shafts and gyroscopes is investigated as well as the motion of rigid bodies filled with fluid, and rocket problems.

Chapter 10 deals with the Hamilton canonical equations of motion using generalized momenta in addition to the generalized coordinates. The properties of canonical transformations result in Jacobi’s theorem for the integral of canonical systems. As an example, the Keplerian motion of a particle in a central force field is chosen. The perturbation theory of Chapter 11 is based on the canonical equations and applied to satellites in the gravitational field of the rotating Earth and to unbalanced heavy tops. Further, the equations for the perturbed motion of a particle are given.

Finally, the variational principles of mechanics are treated in the long Chapter 12. From Hamilton’s action follows Hamilton’s principle denoted as Hamilton-Ostrogradsky principle. Applications to nonholonomic systems and distributed systems like hanging chains and rotating rods are included. The approximate determination of natural frequencies and normal forms is presented for the rotation of a nearly vertical gyroscope. Two appendices on matrix theory and tensor calculus complete the book. A nice list of references as of 1961 and an informative index are also included.

The author succeeded very well with his aim to state problems and to generate the equations of motions of discrete mechanical systems. In particular, the numerous examples show how the methods presented are to be applied to real engineering problems. In contrary to Western literature, only vectors and not matrices are printed in bold characters. Nevertheless, the book is easily readable. The translation is perfect from a science and a language point of view. Therefore, Analytical Mechanics is recommended to graduate students, and to researchers and engineers in industry interested in a rigorous presentation of analytical mechanics. Further, this book should be available in any scientific library.