Singer S.F. Symmetry in Mechanics. A Gentle, Modern Introduction

Birkhauser, Springer Science+Business Media, 2004, 193 pages, ISBN: 0817641459 , 3764341459Recent 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 two-body 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 self-study or classroom use at the undergraduate level. Requires only competency in multivariable calculus, linear algebra and introductory physics.Preliminaries
The Two-Body Problem
Phase Spaces are Symplectic Manifolds
Differential Geometry
Total Energy Functions are Hamiltonian Functions
Symmetries are Lie Group Actions
Infinitesimal Symmetries are Lie Algebras
Conserved Quantities are Momentum Maps
Reduction and The 1\vo-Body Problem