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Classical Mechanics

R. Douglas Gregory
Cambridge University Press
Publication Date: 
Number of Pages: 
[Reviewed by
William J. Satzer
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Classical mechanics has a long and distinguished history, beginning by convention with Newton (or sometimes Galileo) and continuing to the present day. The list of mathematicians who have contributed to the field is also truly illustrious and includes, for example, Gauss, Hamilton, Lagrange, Jacobi, Liouville, Laplace and Kolmogorov. For many years, classical mechanics had a place in the mathematics curriculum, but it has gradually fallen almost exclusively into the domain of physics.

Classical Mechanics is a textbook in analytical mechanics designed for use in an upper level undergraduate course. Most undergraduate physics programs include such a course for their majors. This book very much follows the plan of similar texts such as Classical Dynamics of Particles and Systems by Thornton and Marion or Classical Mechanics: A Modern Perspective by Barger and Olsson. The author’s goal is to cover all the material normally taught in classical mechanics courses from Newton ’s laws to Hamilton ’s equations. Elementary calculus and differential equations are the only stated prerequisites, although some background in mechanics (as from a first year college physics course) would be most desirable. The style is no-nonsense; the author says it is “crisp… there is no waffle!”

Why write another book on classical mechanics so similar in content to other well-known texts? The author never addresses this, but it seems a worthwhile question to ask. It is often dissatisfaction with existing textbooks that drives an ambitious author to write a new one. Students who study classical mechanics generally find it difficult, largely because it poses significant challenges to their ability to integrate mathematical skills and physical reasoning. A relatively high level of hatred for classical mechanics textbooks frequently ensues. When I studied classical mechanics, our textbook seemed to use the words “determinant” and “matrix” interchangeably, and presented an incredibly baroque version of “generalized coordinates”. It is perhaps easy for us to forget how much a bad textbook can magnify a student’s difficulty.

Has this author done better? The writing here is a picture of clarity and directness. The physical layout of the book is attractive. Diagrams and figures are well-drawn. Each page in the book is pleasing to look at. It is amazing how much the physical appearance of a book’s pages can attract or repel the reader’s attention.

The text is divided into four broad sections: Newtonian mechanics of a single particle, multi-particle systems, analytical mechanics (Hamiltonian and Lagrangian formulations) and further topics. The subjects of rotating reference frames and tensors — often especially challenging for students — are deferred to the last section. This is consistent with the author’s overall approach — to concentrate on basic topics to help the student build skills and confidence.

Worked examples and exercises are plentiful and well-chosen. The examples are presented clearly and thoroughly. The exercises include plenty of interesting and challenging problems as well as a modest number of the more routine ones. Solutions to all the exercises are provided.

The author presents a short section on Hohmann transfer orbits that might perhaps lead to a misconception. This subject – on the fuel efficiency for the transit of a vehicle from one orbit to another — is a not often treated in elementary books, though it fits well and provides a good example of motion in an inverse square field. The author presents the basic ideas and then — after saying that proofs in other sources are incomplete — offers a proof of the optimality of the Hohmann transfer orbit. Although his proof is correct — given some assumptions that he doesn’t state — Hohmann orbits are not optimal if one considers full n-body dynamics for n bigger than two. For example, Jerrold Marsden and his colleagues have shown there is a transfer orbit from Ganymede to Europa in the gravitational field of Jupiter with less than half the Hohmann transfer value. Of course, what’s optimal depends on the assumptions, but the assumptions should be clearly stated. Incidentally, Marsden’s methods are being used for trajectory design in NASA’s Genesis mission, the first robotic sample return mission. (See “New methods in celestial mechanics and mission design”, Bull. Amer. Math. Soc. 43 (2006), 43-73.)

This is an attractive and well-written exposition of classical mechanics. I wish it had been my textbook when I was a student.

Bill Satzer ( is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.

 Part I. Newtonian Mechanics of a Single Particle: 1. The algebra and calculus of vectors; 2. Velocity, acceleration and scalar angular velocity; 3. Newton's laws of motion and the law of gravitation; 4. Problems in particle dynamics; 5. Linear oscillations; 6. Energy conservation; 7. Orbits in a central field; 8. Non-linear oscillations and phase space; Part II. Multi-particle Systems: 9. The energy principle; 10. The linear momentum principle; 11. The angular momentum principle; Part III. Analytical mechanics: 12. Lagrange's equations and conservation principle; 13. The calculus of variations and Hamilton's principle; 14. Hamilton's equations and phase space; Part IV. Further Topics: 15. The general theory of small oscillations; 16. Vector angular velocity and rigid body kinematics; 17. Rotating reference frames; 18. Tensor algebra and the inertia tensor; 19. Problems in rigid body dynamics; Appendix. Centres of mass and moments of inertia; Answers to the problems; Bibliography; Index.