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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.

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.