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An Introduction to Hamiltonian Mechanics

Gerardo F. Torres del Castillo
Publication Date: 
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Birkhäuser Advanced Texts
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Steven Deckelman
, on

This book by Gerardo F. Torres del Castillio is a book for beginning students of analytical mechanics, whether they be advanced undergraduates, graduate students or others wishing to learn more about mechanics through self-study. The book does a very nice job of shepherding the reader from Newtonian mechanics to Lagrangian mechanics with an emphasis on Hamiltonian mechanics by means of elementary mathematics that would be familiar to an undergraduate mathematics or physics major. The book makes a deliberate effort to emphasize  intuitive and conceptual understanding and does an exceptionally good job.  The book steers clear of differential geometric language often found in books on mechanics and technicalities unnecessary for beginning students are avoided as well as some of the traditional jargon that has grown up historically in the subject,  e.g. "virtual work'',  "virtual displacement'', "infinitesimal transformation'', "infinitesimal virtual displacement'' etc.  Because of this, as well as the book being introductory, the book is not an encyclopedic tome on analytical mechanics and many topics are not touched upon.  For example only non-relativistic mechanics is considered.

To give an example, the book begins with a simple mechanical system consisting of a frictionless block on a frictionless wedge. Newton's laws are used to find the equations of motion. After constraint forces are removed, D'Alembert's principle is easily derived. This then leads in a natural way to the Lagrange equations and the Lagrangian. Numerous worked examples and exercises are given. The book then moves into rigid body mechanics, again with further applications of the Lagrangian. This then sets the stage for Hamilton's equations and mechanics.   The final chapters give a like introduction to canonical transformations and the Hamilton-Jacobi equation. The very last section gives some further exercise applications in geometrical optics.

A particularly helpful aspect of the book is that all of the exercises have detailed solutions given in the back of the book making it ideal for self-study. In chapter three on rigid bodies the Einstein notation is introduced in tensor calculations without mention, but the examples are very illuminating and would be presumably familiar to anyone with a modicum of background in mathematics. This book is a must-have library book for any mathematics or physics library.

Steven Deckelman is a professor of mathematics at the University of Wisconsin-Stout, where he has been since 1997. He received his Ph.D from the University of Wisconsin-Madison in 1994 for a thesis in several complex variables written under Patrick Ahern. Some of his interests include complex analysis, mathematical biology and the history of mathematics.

See the table of contents in the publisher's webpage.