A Primer of Analytical Mechanics

This is an introductory text on analytical mechanics prepared by the author for a half-semester course with an audience of undergraduates “not theoretically and mathematically minded” who were largely interested in experimental physics. It is an ambitious book for such an audience. The contents require more than a little mathematical sophistication, and the pace is relatively fast.

The author notes that analytical mechanics is sometimes highly regarded because of its formal structure, variational principles, and general mathematical elegance. He wishes instead to emphasize its strong physical motivations and its ability to offer the simplest approach to a large class of physical problems.

He begins with Newtonian mechanics and observes that it has deficiencies. In Cartesian coordinates, the variables are often not independent, equations in Newtonian form do not have simple transformation properties under changes of coordinates, and fictitious forces arise in non-inertial frames of reference.

Many of these difficulties are resolved in Lagrange’s formulation of mechanics. The author argues that Lagrange’s equations provide the most economical description of the time evolution of a system, incorporate constraints efficiently, and have the advantage of covariance under changes of coordinates. He gives examples of the Foucault pendulum, of the computation of the eastward displacement of a falling object, and of the motion of a charged particle in an electromagnetic field.

The author then turns to the Hamiltonian formulation of mechanics, Hamilton’s equations and canonical transformations that preserve the form of these equations. Pretty quickly he moves on to describe Poisson brackets and their use in identifying constants of the motion for a dynamical system. Then he turns to generating functions for canonical transformations and touches briefly on Noether’s famous theorem that relates symmetries and conservation laws.

A final chapter takes up the advanced topic of a common Poisson algebra for classical and quantum mechanics. (The author suggests that Dirac’s proposal for this algebra has mathematical inconsistencies, and offers an alternative.)

All this is in about one hundred pages. It is an odd book, particularly for the audience that the author wishes to address. It begins at a level that would perhaps be a little uncomfortable for that audience and raises the level throughout the book. Although he calls the book a primer, part of the author’s aim is to raise questions about prior treatments of mechanics and to make a case for the superiority of the Lagrangian formulation. Most of this is likely to sail right over the heads of his intended readers.

The book has an appendix with six problems and their solutions using a Lagrangian formulation. It is the clearest and probably the most useful part of the book.

Anyone looking for a good introduction to analytical mechanics might start with Goldstein’s Classical Mechanics, a well-written classic aimed primarily at physics students. Those interested in a mathematician’s take on mechanics should consider Arnóld’s Mathematical Methods of Classical Mechanics or Abraham and Marsden’s Foundations of Mechanics.

Bill Satzer (bsatzer@gmail.com) was a senior intellectual property scientist at 3M Company. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.