Symmetry in Mechanics : A Gentle Modern Introduction And what is the use thought Alice of a book without pictures or conversations in it Lewis Carroll This book is written for modem undergraduate students not the ideal stu­ dents that mathematics professors wish for (and who occasionally grace our campuses) but the students like many the author has taught: talented but ap­ preciating review and reinforcement of past course work; willing to work hard but demanding context and motivation for the mathematics they are learning. To suit this audience the author eschews density of topics and efficiency of presentation in favor of a gentler tone a coherent story digressions on mathe­ maticians physicists and their notations simple examples worked out in detail and reinforcement of the basics. Dense and efficient texts play a crucial role in the education of budding (and budded) mathematicians and physicists. This book does not presume to improve on the classics in that genre. Rather it aims to provide those classics with a large new generation of appreciative readers. This text introduces some basic constructs of modern symplectic geometry in the context of an old celestial mechanics problem the two body problem. We present the derivation of Keplers laws of planetary motion from Newtons laws of gravitation first in the style of an undergraduate physics course and x Preface then again in the language of symplectic geometry. No previous exposure to symplectic geometry is required: we introduce and illustrate all necessary con­ structs. 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.