Olver P.J. Equivalence, Invariants, and Symmetry

Cambridge University Press, 1995. — 525 p. — ISBN: 0-521-47811-1.Drawing on a wide range of mathematical disciplines, including geometry, analysis, applied mathematics and algebra, this book presents an innovative synthesis of methods used to study problems of equivalence and symmetry which arise in a variety of mathematical fields and physical applications. Systematic and constructive methods for solving equivalence problems and calculating symmetries are developed and applied to a wide variety of mathematical systems, including differential equations, variational problems, manifolds, Riemannian metrics, polynomials and differential operators. Particular emphasis is given to the construction and classification of invariants, and to the reductions of complicated objects to simple canonical forms. This book will be a valuable resource for students and researchers in geometry, analysis, algebra, mathematical physics and other related fields.Geometric Foundations
Lie Groups
Representation Theory
Jets and Contact Transformations
Differential Invariants
Symmetries of Differential Equations
Symmetries of Variational Problems
Equivalence of Coframes
Formulation of Equivalence Problems
Cartan's Equivalence Method
Involution
Prolongation of Equivalence Problems
Differential Systems
Frobenius' Theorem
The Cartan-Kähler Existence Theorem