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Symmetry in Mechanicsby Stephanie F. Singer
Synopses & ReviewsPublisher Comments:"Symmetry in Mechanics is directed to students at the undergraduate level and beyond, and offers a lovely presentation of the subject...The first chapter presents a standard derivation of the equations for twobody planetary motion. Kepler's laws are then obtained and the rule of conservation laws is emphasized. ..... Singer uses this example from classical physics throughout the book as a vehicle for explaining the concepts of differential geometry and for illustrating their use. ... These ideas and techniques will allow the reader to understand advanced texts and research literature in which considerably more difficult problems are treated and solved by identical or related methods. The book contains 122 student exercises, many of which are solved in an appendix. The solutions, especially, are valuable for showing how a mathematician approaches and solves specific problems. Using this presentation, the book removes some of the language barriers that divide the worlds of mathematics and physics..."  Physics Today
Recent years have seen the appearance of several books bridging the gap between mathematics and physics; most are aimed at the graduate level and above. Symmetry in Mechanics: A Gentle, Modern Introduction is aimed at anyone who has observed that symmetry yields simplification and wants to know why. The monograph was written with two goals in mind: to chip away at the language barrier between physicists and mathematicians and to link the abstract constructions of symplectic mechanics to concrete, explicitly calculated examples. The context is the twobody problem, i.e., the derivation of Kepler's Laws of planetary motion from Newton's laws of gravitation. After a straightforward and elementary presentation of this derivation in the language of vector calculus, subsequent chapters slowly and carefully introduce symplectic manifolds, Hamiltonian flows, Lie group actions, Lie algebras, momentum maps and symplectic reduction, with many examples, illustrations and exercises. The work ends with the derivation it started with, but in the more sophisticated language of symplectic and differential geometry. For the student, mathematician or physicist, this gentle introduction to mechanics via symplectic reduction will be a rewarding experience. The freestanding chapter on differential geometry will be a useful supplement to any first course on manifolds. The book contains a number of exercises with solutions, and is an excellent resource for selfstudy or classroom use at the undergraduate level. Requires only competency in multivariable calculus, linear algebra and introductory physics. Synopsis:This book presents a straightforward introduction to mechanics via symplectic reduction. Following a short historical introduction to the subject, the author gives a careful exposition of the twobody problem using it as a vehicle to link the abstract constructions of symplectic and differential geometry to concrete, explicitly calculated examples. Accessible to those with only a background in multivariable calculus and linear algebra and presenting many examples, illustrations and exercises, the book will be an excellent selfstudy resource.
Table of ContentsPreface * Preliminaries * 1. The TwoBody Problem * 2. Phase Spaces are Symplectic Manifolds * 3. Differential Geometry (Optional) * 4. Total Energy Functions are Hamiltonian Functions * 5. Symmetries are Lie Group Actions * 6. Infinitesimal Symmetries are Lie Algebras * 7. Conserved Quantities are Momentum Maps * 8. Reduction and the TwoBody Problem * Solutions to Exercises * Bibliography * Index
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