A Practical Guide to the Invariant Calculus

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.

Elizabeth Louise Mansfield is Professor of Mathematics at the University of Kent, Canterbury

Preface	p. ix
Introduction to invariant and equivariant problems	p. 1
The curve completion problem	p. 1
Curvature flows and the Korteweg-de Vries equation	p. 4
The essential simplicity of the main idea	p. 5
Overview of this book	p. 9
How to read this book...	p. 11
Actions galore	p. 12
Introductory examples	p. 12
Actions	p. 18
Semi-direct products	p. 23
New actions from old	p. 24
Induced actions on functions	p. 24
Induced actions on products	p. 25
Induced actions on curves	p. 26
Induced action on derivatives: the prolonged action	p. 27
Some typical group actions in geometry and algebra	p. 31
Properties of action	p. 33
One parameter Lie groups	p. 37
The infinitesimal vector fields	p. 39
The prolongation formula	p. 44
From infinitesimals to actions	p. 46
Calculus on Lie group	p. 51
Local coordinates	p. 51
Tangent vectors on Lie groups	p. 55
Tangent vectors for matrix Lie groups	p. 58
Some standard notations for vectors and tangent maps in coordinates	p. 60
Vector fields and integral curves	p. 62
Integral curves in terms of the exponential of a vector field	p. 66
Tangent vectors at the identity versus one parameter subgroups	p. 67
The exponential map	p. 68
Associated concepts for transformation groups	p. 69
From Lie group to Lie algebra	p. 73
The Lie bracket of two vector fields on Rⁿ	p. 74
Frobenius' Theorem	p. 82
The Lie algebra bracket on T_eG	p. 87
The Lie algebra bracket for matrix Lie groups	p. 90
The Lie algebra bracket for transformation groups, and Lie's Three Theorems	p. 95
The Adjoint and adjoint actions for transformation groups	p. 105
Moving frames	p. 114
Moving frames	p. 114
Transversality and the converse to Theorem 4.1.3	p. 122
Frames for SL(2) actions	p. 126
Invariants	p. 127
Invariant differentiation	p. 132
Invariant differentiation for linear actions of matrix Lie groups	p. 139
*Recursive construction of frames	p. 140
*Joint invariants	p. 148
On syzygies and curvature matrices	p. 151
Computations with differential invariants	p. 152
Syzygies	p. 159
Curvature matrices	p. 161
Notes for symbolic computation	p. 167
*The Serret-Frenet frame	p. 168
*Curvature matrices for linear actions	p. 175
*Curvature flows	p. 180
Invariant ordinary differential equations	p. 185
The symmetry group of an ordinary differential equation	p. 187
Solving invariant ordinary differential equations using moving frames	p. 189
First order ordinary differential equations	p. 192
SL(2) invariant ordinary differential equations	p. 195
Schwarz' Theorem	p. 195
The Chazy equation	p. 197
Equations with solvable symmetry groups	p. 199
Notes on symbolic and numeric computation	p. 202
Using only the infinitesimal vector fields	p. 202
Variational problems with symmetry	p. 206
Introduction to the Calculus of Variations	p. 206
Results and non-results for Lagrangians involving curvature	p. 212
Group actions on Lagrangians and Noether's First Theorem	p. 216
Moving frames and Noether's Theorem, the appetizer	p. 220
Calculating invariantised Euler-Lagrange equations directly	p. 222
The case of invariant, unconstrained independent variables	p. 224
The case of non-invariant independent variables	p. 227
The case of constrained independent variables such as arc length	p. 230
The 'mumbo jumbo' -free rigid body	p. 232
Moving frames and Noether's Theorem, the main course	p. 236
References	p. 241
Index	p. 244
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A Practical Guide to the Invariant Calculus

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