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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. {\it Symmetry in Mechanics: A Gentle, Modern Introduction} is geared towards a broad audience, requiring only competency in multivariable calculus, linear algebra, and introductory physics.
This work 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 to concrete, explicitly calculated examples. The context is a careful exposition of the two-body problem, namely, the derivation of Kepler's laws of planetary motion from Newton's laws of gravitation.
Key features of this work include:
· straightforward and elementary presentation of the derivation of Kepler's laws in the language of vector calculus,
· short historical introduction to the subject,
· gentle introduction to symplectic manifolds, Hamiltonian flows, Lie group actions, Lie algebras, momentum maps, and symplectic reduction with many examples, exercises, and solutions,
· cyclical treatment of material---book ends with the derivation it started with, in the language of symplectic and differential geometry.
For the student, mathematician or physicist who has noticed that symmetry yields simplification and wants to know why, this book will be a rewarding experience. The book is an excellent resource for self-study or classroom use at the undergraduate level, requiring only competency in multivariable calculus, linear algebra and introductory physics.
Rezensionen / Stimmen
"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 two-body 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
"This is a very interesting book. Those educated in traditional mechanics will acquire [from reading it] knowledge of modern mathematics hidden beyond traditional concepts in the realm of celestial mechanics, [and] . . . pure mathematicians will understand how their discipline enters into practical problems. The author shows how fundamental concepts of symplectic geometry implicitly occur in mechanics . . . the mathematical presentation is ingenious and subtle. There are a lot of exercises for the reader and the solutions of most of them are given in a separate chapter. I can highly recommend this book to undergraduate and PhD students . . . it is ideally suited for teaching a course on the subject."
-Mathematical Reviews
Auflage
Sprache
Verlagsort
Zielgruppe
Für höhere Schule und Studium
Für Beruf und Forschung
Research
Illustrationen
4
4 s/w Abbildungen
XII, 193 p. 4 illus.
Maße
Höhe: 235 mm
Breite: 155 mm
Dicke: 12 mm
Gewicht
ISBN-13
978-0-8176-4145-0 (9780817641450)
DOI
10.1007/978-1-4612-0189-2
Schweitzer Klassifikation
0 Preliminaries.- 1 The Two-Body Problem.- 2 Phase Spaces are Symplectic Manifolds.- 3 Differential Geometry.- 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 Two-Body Problem.- Recommended Reading.- Solutions.- References.
