# Symmetry in Mechanics

## A Gentle, Modern Introduction

• Stephanie Frank Singer
Textbook

1. Front Matter
Pages i-xiii
2. Stephanie Frank Singer
Pages 1-4
3. Stephanie Frank Singer
Pages 5-18
4. Stephanie Frank Singer
Pages 19-45
5. Stephanie Frank Singer
Pages 47-63
6. Stephanie Frank Singer
Pages 65-82
7. Stephanie Frank Singer
Pages 83-100
8. Stephanie Frank Singer
Pages 101-112
9. Stephanie Frank Singer
Pages 113-122
10. Stephanie Frank Singer
Pages 123-139
11. Back Matter
Pages 141-193

### 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 Kepler's laws of planetary motion from Newton's 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.

### Keywords

Lie group Lie groups algebra classical mechanics differential geometry ksa linear algebra manifold symplectic geometry sympletic geometry

#### Authors and affiliations

• Stephanie Frank Singer
• 1