Customer Reviews

Perfect Form

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This is an engaging book, written on a fairly basic level. Any junior with some calculus should be able to handle it. The author has done a great job of introducing the calculus of variations, Lagrange multipliers, etc, and applying them to clear examples from physics (Fermat's principle, Lagrangians and Hamiltonians). I only wish he had expanded the topics somewhat to introduce a few more topics to whet the appetite, such as phase spaces, Liouville's theorem, Noether's theorem.

I just took an independent study in the calculus of variations out of Gelfand's classic text. I covered the first four chapters which is a nice introduction. However the text is pretty technical and so Perfect Form (PF) was a great companion. Its laid back, accessible to a sophomore physics student and fine for self study. It has a range of physical problems from calculations to nice little problems to think about.

Moreover, it motivates the material well. This is one of those books that keeps driving home a few, just a few points and avoids too many topics. For instance, I was never knew why the lagrangian should be the difference of kinetic and potential energies, this book will motivate this form.

Finally, its a realistic book. I found no great effort in reading the entire book and working about 3/4 of the problems (some I just didn't find interesting) on my own in a busy semester. This is just a fun little book that shows you some variational methods!

For a third or fourth year student in physics this short book, Perfect Form, would be near perfect as either a short overview of variational methods, or as a supplementary text for an advanced classical physics course.

I have occasionally encountered variational methods, but until reading Perfect Form I had not appreciated the significance and scope and even fascination of this topic. In a little more than one hundred pages Dr. Don Lemons does a credible job of introducing a wide range of physics problems amenable to variational methods.

He begins with optics and Fermat's Principle of Least Time and thereby motivates the derivation of the Euler-Lagrange equation. In later chapters he examines the principle of least potential energy, Lagrange multipliers, the principle of least action, and Hamilton's principle, in both a restricted and more general form. The supplementary problems at the end of each chapter are few in number, but are carefully defined and are more like tutorials than standalone problems.

In my experience textbooks dedicated to this topic - like Calculus of Variations by Robert Weinstock and Introduction to the Calculus of Variations by Hans Sagan - are difficult and require considerable mathematical maturity. Other texts - like Advanced Calculus of Several Variables (C. H. Edwards) and Advanced Mathematical Methods for Engineering and Science Students (Stephenson and Radmore) and Mathematics Applied to Continuum Mechanics (L. A. Segel) - often relegate this subject to a single (and often final) chapter.

Most undergraduates are unlikely to have time for a formal course in calculus of variations. With this book Don Lemons has convinced me that this topic is too important and too interesting to ignore. I recommend that you acquire a copy of Perfect Form for self-study or as supplementary text.