Contents

Chapter 1. Introduction 1

1.1. Background 1

1.2. The Main Results of This Paper. 3

1.3. Future Work and Related Issues. 4

Chapter 2. Preliminary Constructions 7

2.1. Notation and general assumptions. 7

2.2. Connections on Principal Bundles. 8

2.3. Associated Bundles. 10

2.4. The Bundles TQ/G and T(Q/G) © g 14

Chapter 3. The Lagrange-Poincare Equations 17

3.1. The Geometry of Variations 17

3.2. The Euler-Lagrange and Euler-Poincare Operators 21

3.3. The Lagrange-Poincare Operator 28

3.4. The Reduced Variational Principle 34

Chapter 4. Wong's Equations and Coordinate Formulas 37

4.1. Wong's Equations 37

4.2. The Local Vertical and Horizontal Equations 38

Chapter 5. The Lie Algebra Structure on Sections of the Reduced Bundle 43

5.1. The Bundle T(Q/G) © g Revisited 43

5.2. The Lie Algebra of Sections of T(Q/G) 0 0 45

Chapter 6. Reduced Tangent Bundles 51

6.1. The Geometry of Lagrange-Poincare Bundles 52

6.2. Reduction of Lagrange-Poincare Bundles 61

6.3. Reduction by Stages of objects of £*P 71

6.4. The Subcategory *HX and Reduction by Stages of Variational

Principles on TQ. 78

Chapter 7. Further Examples 85

7.1. Semidirect Products 85

7.2. Central Extensions 86

7.3. Rigid Body with Rotors 88

7.4. Systems Depending on a Parameter 88

Chapter 8. The Category £*$* and Poisson Geometry 95

8.1. The Poisson Bracket on Duals of Objects of £*£ 95