Dynamical Symmetry

Description

Whenever systems are governed by continuous chains of causes and effects, their behavior exhibits the consequences of dynamical symmetries, many of them far from obvious. * Dynamical Symmetry* introduces the reader to Sophus Lie's discoveries of the connections between differential equations and continuous groups that underlie this observation. It develops and applies the mathematical relations between dynamics and geometry that result. Systematic methods for uncovering dynamical symmetries are described, and put to use. Much material in the book is new and some has only recently appeared in research journals.

Though Lie groups play a key role in elementary particle physics, their connection with differential equations is more often exploited in applied mathematics and engineering. * Dynamical Symmetry* bridges this gap in a novel manner designed to help readers establish new connections in their own areas of interest. Emphasis is placed on applications to physics and chemistry. Applications to many of the other sciences illustrate both general principles and the ubiquitousness of dynamical symmetries.

Readership: Established physicists, atomic physicists, molecular physicists, theoretical chemists, biophysicists, applied mathematicians; junior and senior undergraduate majors in physics, mathematics, physical chemistry; graduate students in these areas, and in chemical physics, quantum chemistry, biophysics.

This book provides a gentle introduction to Lie symmetries. The natural audience for this book will consist primarily of physicists, and the way the book has been written ensures that most of the material can be readily understood with a modest mathematical background. Applications to classical and quantum mechanics are emphasized throughout.
The text begins with general considerations about symmetries, before moving to Lie transformation groups and Lie algebras. This roughly constitutes the first half of the book. The second half deals with applications of the general theory to Hamiltonian dynamical systems (e.g., the Kepler problem), to quantum mechanics (e.g., spectrum-generating algebras and analysis of simple atomic and molecular systems), and to the Maxwell equations. Many concrete examples are analyzed in detail.

Reviewed by Alberto Enciso