Classical dynamics. A contemporary approach.

*(English)*Zbl 0918.70001
Cambridge: Cambridge University Press. xxvi, 670 p. (1998).

In recent years two circumstances have contributed to the advance of mechanics: by the one hand, the use of qualitative geometric and topological ideas instead of the traditional analytical ones, and, by the other hand, the use of computers have permitted a finer quantitative analysis of dynamical systems. The purpose of the authors is to present both aspects, as well as the classical approach to mechanics, in a unique book. And, indeed, it is a book on mechanics which differs from the books in the existing literature.

Here is a brief description of chapters. Chapter 1 is devoted to a review of Newton’s equations of motion, and of the basic concepts in Newton mechanics, that is, the configuration spaces are Euclidean. The Lagrangian formalism is introduced in chapter 2; here we are in a more general setting, and the configuration spaces are differentiable manifolds. The authors introduce generalized coordinates and derive the Euler-Lagrange equations from Newton’s law. In fact, the introduction of generalized variables allows constraint forces to be eliminated. The fundamental concept of Lagrangian function is discussed, as well as the invariance of the Euler-Lagrange equations (indeed, a consequence of the symplectic formalism). Additionally, the authors discuss in chapter 2 central forces, and at the end introduce tangent bundles (spaces of velocities).

Chapter 3 is devoted to Lagrangian dynamics, where the Euler-Lagrange equations are derived from a variational principle (Hamilton’s principle). The authors also consider mechanical systems subjected to constraints, and derive the vakonomic and nonholonomic equations of motion. The relation between symmetries and conserved quantities is established via the Noether theorem (only point symmetries are considered for simplicity). Finally, the authors discuss dissipative and driving forces, and like in other chapters, some fundamental concepts in the differential calculus on manifolds are introduced in a very natural way.

Chapter 4 deals with a completely different topic, scattering. However, this material will be useful in subsequent chapters, mainly on nonlinear systems, perturbation theory, and field theory. An introduction to Hamiltonian mechanics is given in chapter 5, and some fundamental concepts like symplectic forms, Hamiltonian flows and Poisson brackets are introduced. Both formulations, Lagrangian and Hamiltonian, are related by the Legendre transformation. The authors discuss Darboux and Liouville theorems, and some parts of Hamilton-Jacobi theory, but this topic will be one of the main subjects of the next chapter. Indeed, in chapter 6 the authors discuss integrability of dynamical systems: separability, completely integrable systems and action-angle variables, and some elements of perturbation theory. In chapter 7 they study nonlinear dynamical systems, focussing on some general topics, like stability of solutions. Analytical as well as numerical approximations are given. Then chaotic systems are introduced, and an idea of the proof of KAM theorem is given. Chapter 8 is devoted to a classical topic, rigid bodies, and here Lagrangian and Hamiltonian formulations are extensively applied. The book ends with a brief introduction to continuum mechanics. To be precise, this is an introduction to classical field theory. The geometric structures underlying these theories are not treated here, but they are out of the scope of this book.

Summing up, this is a really nice book for graduate (and even undergraduate) students in many fields like physics, mathematics, and engineering.

Here is a brief description of chapters. Chapter 1 is devoted to a review of Newton’s equations of motion, and of the basic concepts in Newton mechanics, that is, the configuration spaces are Euclidean. The Lagrangian formalism is introduced in chapter 2; here we are in a more general setting, and the configuration spaces are differentiable manifolds. The authors introduce generalized coordinates and derive the Euler-Lagrange equations from Newton’s law. In fact, the introduction of generalized variables allows constraint forces to be eliminated. The fundamental concept of Lagrangian function is discussed, as well as the invariance of the Euler-Lagrange equations (indeed, a consequence of the symplectic formalism). Additionally, the authors discuss in chapter 2 central forces, and at the end introduce tangent bundles (spaces of velocities).

Chapter 3 is devoted to Lagrangian dynamics, where the Euler-Lagrange equations are derived from a variational principle (Hamilton’s principle). The authors also consider mechanical systems subjected to constraints, and derive the vakonomic and nonholonomic equations of motion. The relation between symmetries and conserved quantities is established via the Noether theorem (only point symmetries are considered for simplicity). Finally, the authors discuss dissipative and driving forces, and like in other chapters, some fundamental concepts in the differential calculus on manifolds are introduced in a very natural way.

Chapter 4 deals with a completely different topic, scattering. However, this material will be useful in subsequent chapters, mainly on nonlinear systems, perturbation theory, and field theory. An introduction to Hamiltonian mechanics is given in chapter 5, and some fundamental concepts like symplectic forms, Hamiltonian flows and Poisson brackets are introduced. Both formulations, Lagrangian and Hamiltonian, are related by the Legendre transformation. The authors discuss Darboux and Liouville theorems, and some parts of Hamilton-Jacobi theory, but this topic will be one of the main subjects of the next chapter. Indeed, in chapter 6 the authors discuss integrability of dynamical systems: separability, completely integrable systems and action-angle variables, and some elements of perturbation theory. In chapter 7 they study nonlinear dynamical systems, focussing on some general topics, like stability of solutions. Analytical as well as numerical approximations are given. Then chaotic systems are introduced, and an idea of the proof of KAM theorem is given. Chapter 8 is devoted to a classical topic, rigid bodies, and here Lagrangian and Hamiltonian formulations are extensively applied. The book ends with a brief introduction to continuum mechanics. To be precise, this is an introduction to classical field theory. The geometric structures underlying these theories are not treated here, but they are out of the scope of this book.

Summing up, this is a really nice book for graduate (and even undergraduate) students in many fields like physics, mathematics, and engineering.

Reviewer: M.de León (Madrid)

##### MSC:

70-02 | Research exposition (monographs, survey articles) pertaining to mechanics of particles and systems |

70Hxx | Hamiltonian and Lagrangian mechanics |

37Cxx | Smooth dynamical systems: general theory |

70Fxx | Dynamics of a system of particles, including celestial mechanics |

70Gxx | General models, approaches, and methods in mechanics of particles and systems |