Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgments
- Introduction
- 1 Geometric Foundations
- 2 Lie Groups
- 3 Representation Theory
- 4 Jets and Contact Transformations
- 5 Differential Invariants
- 6 Symmetries of Differential Equations
- 7 Symmetries of Variational Problems
- 8 Equivalence of Coframes
- 9 Formulation of Equivalence Problems
- 10 Cartan's Equivalence Method
- 11 Involution
- 12 Prolongation of Equivalence Problems
- 13 Differential Systems
- 14 Frobenius' Theorem
- 15 The Cartan–Kähler Existence Theorem
- Tables
- References
- Symbol Index
- Author Index
- Subject Index
Introduction
Published online by Cambridge University Press: 05 August 2012
- Frontmatter
- Contents
- Preface
- Acknowledgments
- Introduction
- 1 Geometric Foundations
- 2 Lie Groups
- 3 Representation Theory
- 4 Jets and Contact Transformations
- 5 Differential Invariants
- 6 Symmetries of Differential Equations
- 7 Symmetries of Variational Problems
- 8 Equivalence of Coframes
- 9 Formulation of Equivalence Problems
- 10 Cartan's Equivalence Method
- 11 Involution
- 12 Prolongation of Equivalence Problems
- 13 Differential Systems
- 14 Frobenius' Theorem
- 15 The Cartan–Kähler Existence Theorem
- Tables
- References
- Symbol Index
- Author Index
- Subject Index
Summary
The most basic of the themes to be discussed is the concept of equivalence. The fundamental equivalence problem is to determine whether two geometric objects can be transformed into each other by a suitable change of variables. For example, one might be interested in whether two given differential equations are, in fact, the same equation rewritten in terms of different independent and dependent variables. As a particular instance, the linearization problem is to determine whether a differential equation can be mapped to a linear differential equation via a change of variables. In this manner, a solution to the general equivalence problem for differential equations will include, as a very particular case, the characterization of all linearizable differential equations. As a second example, in the calculus of variations, the basic equivalence problem is to determine when two variational problems can be identified under a change of variables. This problem forms a subcase of the differential equation equivalence problem, since the change of variables will also map their Euler-Lagrange equations to each other. In algebra, the problem of equivalence of (homogeneous) polynomials under linear transformations also falls under our broad purview. Geometric equivalence problems lie at the foundations of Riemannian geometry, where one wants to know whether two given Riemannian manifolds can be identified under a (local) diffeomorphism, and, similarly, in symplectic geometry, where one asks the same question for two symplectic manifolds.
The determination of the symmetry group of a geometric object can be regarded as a special case of the general equivalence problem. Indeed, provided it lies in the admissible class of changes of variables, a symmetry is merely a self-equivalence of the object.
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- Equivalence, Invariants and Symmetry , pp. 1 - 6Publisher: Cambridge University PressPrint publication year: 1995