This book explains recent results in the theory of moving frames that concern the symbolic manipulation of invariants of Lie group actions. In particular, theorems concerning the calculation of generators of algebras of differential invariants, and the relations they satisfy, are discussed in detail. The author demonstrates how new ideas lead to significant progress in two main applications: the solution of invariant ordinary differential equations and the structure of Euler-Lagrange equations and conservation laws of variational problems. The expository language used here is primarily that of undergraduate calculus rather than differential geometry, making the topic more accessible to a student audience. More sophisticated ideas from differential topology and Lie theory are explained from scratch using illustrative examples and exercises. This book is ideal for graduate students and researchers working in differential equations, symbolic computation, applications of Lie groups and, to a lesser extent, differential geometry.
Author(s): Elizabeth Louise Mansfield
Series: Cambridge Monographs on Applied and Computational Mathematics
Publisher: CUP
Year: 2010
Language: English
Pages: 261
Cover......Page 1
Half-title......Page 3
Series-title......Page 4
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface......Page 11
The curve completion problem......Page 15
Curvature flows and the Korteweg-de Vries equation......Page 18
The essential simplicity of the main idea......Page 19
Overview of this book......Page 23
How to read this book …......Page 25
1.1 Introductory examples......Page 26
1.2 Actions......Page 32
1.2.1 Semi-direct products......Page 37
1.3.1 Induced actions on functions......Page 38
1.3.2 Induced actions on products......Page 39
1.3.3 Induced actions on curves......Page 40
1.3.4 Induced action on derivatives: the prolonged action......Page 41
1.3.5 Some typical group actions in geometry and algebra......Page 45
1.4 Properties of actions......Page 47
1.5 One parameter Lie groups......Page 51
1.6 The infinitesimal vector fields......Page 53
1.6.1 The prolongation formula......Page 58
1.6.2 From infinitesimals to actions......Page 60
2.1 Local coordinates......Page 65
2.2 Tangent vectors on Lie groups......Page 69
2.2.1 Tangent vectors for matrix Lie groups......Page 72
2.2.2 Some standard notations for vectors and tangent maps in coordinates......Page 74
2.3 Vector fields and integral curves......Page 76
2.3.1 Integral curves in terms of the exponential of a vector field......Page 80
2.4 Tangent vectors at the identity versus one parameter subgroups......Page 81
2.5 The exponential map......Page 82
2.6 Associated concepts for transformation groups......Page 83
3 From Lie group to Lie algebra......Page 87
3.1 The Lie bracket of two vector fields on Rn......Page 88
3.1.1 Frobenius' Theorem......Page 96
3.2 The Lie algebra bracket on TeG......Page 101
3.2.1 The Lie algebra bracket for matrix Lie groups......Page 104
3.2.2 The Lie algebra bracket for transformation groups, and Lie's Three Theorems......Page 109
3.2.2.1 Prolongations of Lie algebras of vector fields......Page 113
3.2.2.2 An important formula......Page 114
3.2.2.3 Lie's First Theorem......Page 117
3.3 The Adjoint and adjoint actions for transformation groups......Page 119
4.1 Moving frames......Page 128
4.2 Transversality and the converse to Theorem 4.1.3......Page 136
4.3 Frames for SL(2) actions......Page 140
4.4 Invariants......Page 141
4.5 Invariant differentiation......Page 146
4.5.1 Invariant differentiation for linear actions of matrix Lie groups......Page 153
4.6 Recursive construction of frames......Page 154
4.7 Joint invariants......Page 162
5 On syzygies and curvature matrices......Page 165
5.1 Computations with differential invariants......Page 166
5.1.1 Syzygies......Page 173
5.2 Curvature matrices......Page 175
5.3 Notes for symbolic computation......Page 181
5.4 The Serret-Frenet frame......Page 182
5.5 Curvature matrices for linear actions......Page 189
5.6 Curvature flows......Page 194
6 Invariant ordinary differential equations......Page 199
6.1 The symmetry group of an ordinary differential equation......Page 201
6.2 Solving invariant ordinary differential equations using moving frames......Page 203
Step 0......Page 204
Step 4......Page 205
6.3 First order ordinary differential equations......Page 206
6.4.1 Schwarz' Theorem......Page 209
6.4.2 The Chazy equation......Page 211
6.5 Equations with solvable symmetry groups......Page 213
6.7 Using only the infinitesimal vector fields......Page 216
7.1 Introduction to the Calculus of Variations......Page 220
7.1.1 Results and non-results for Lagrangians involving curvature......Page 226
7.2 Group actions on Lagrangians and Noether’s First Theorem......Page 230
7.2.1 Moving frames and Noether's Theorem, the appetizer......Page 234
7.3 Calculating invariantised Euler–Lagrange equations directly......Page 236
7.3.1 The case of invariant, unconstrained independent variables......Page 238
7.3.2 The case of non-invariant independent variables......Page 241
7.3.3 The case of constrained independent variables such as arc length......Page 244
7.3.4 The 'mumbo jumbo'-free rigid body......Page 246
7.4 Moving frames and Noether’s Theorem, the main course......Page 250
References......Page 255
Index......Page 258
