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Classical Mechanics: Theory and Mathematical Modeling

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This volume is based on lectures given by the author on rational mechanics at the University of Rome over a period of more than a decade. The author has developed an approach to the subject that has a distinct flavor and a style of its own. It emphasizes a rigorous mathematical approach to the modeling of mechanical systems and largely avoids modern abstract methods.

The topics included here are common to most treatments of classical mechanics at the advanced undergraduate or beginning graduate level: kinematics and dynamics of point masses, constrained systems, Lagrange’s equations, rigid body motion, precession and gyroscopic motion, variational principles, the Hamilton-Jacobi equations and canonical coordinates. Where it differs from other texts is in its inclusion of a variety of more specialized topics. For example, the author’s treatment of the n-body problem includes a discussion of total collapse of the n-body system and the associated Sundman inequality. This is a topic usually discussed only in books aimed at specialists. The author’s treatment includes a concluding chapter on fluid dynamics, a subject that is also commonly treated separately from classical mechanics.

No mention is made anywhere of manifolds, tangent and cotangent bundles, or other tools from differential geometry that are used in more abstract and global treatments. The author works exclusively locally, in coordinates, and devotes a fair amount of attention to several of the standard coordinate systems used by physicists — cylindrical, elliptic, parabolic — and equations of motion in those systems. The treatment is consistently rigorous and traditional, and probably aligned (in spirit if not in rigor) with the way classical mechanics is treated in most graduate physics programs.

The author packs quite a bit into this volume; that tends to make the book more useful as a reference than as a potential textbook.  There are very few examples that are not simply further elaborations of the theory. Neither will the reader find any numerical calculations or exercises asking for numerical solutions. This overwhelming emphasis on theory without even a nod toward practice and applications seems odd for a subject that has natural and intuitive connections to the physical world. It makes the subject seem dry and rather lifeless.

The author assumes that the reader has a background that includes linear algebra, multivariable calculus, the basic theory of differential equations and elementary physics. His target audience includes mathematics, physics and engineering students at advanced undergraduate or beginning graduate level.

Each chapter has a “Problems and Complements” section that includes exercises and additional exposition related to the contents of that chapter. Complements are not distinguished from Problems; only the occasional request to prove or show signals the difference. There are quite a few more Complements than Problems. Anyone considering using this as a textbook would need to provide supplementary exercises.

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.

Date Received: 
Monday, November 8, 2010
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Emmanuele DiBenedetto
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William J. Satzer
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Tuesday, August 16, 2011