Transformation Groups Applied to Mathematical Physics (Hardcover, 1985)

I: Point Transformations.- Introductory Chapter: Group and Differential Equations.- 1. Continuous groups.- 1.1 Topological groups.- 1.2 Lie groups.- 1.3 Local groups.- 1.4 Local Lie groups.- 2. Lie algebras.- 2.1 Definitions.- 2.2 Lie algebras and local Lie groups.- 2.3 Inner automorphisms.- 2.4 The Levi-Mal'cev theorem.- 3. Transformation groups.- 3.1 Local transformation groups.- 3.2 Lie's equation.- 3.3 Invariants.- 3.4 Invariant manifolds.- 4. Invariant differential equations.- 4.1 Prolongation of point transformations.- 4.2 The defining equation.- 4.3 Invariant and partially invariant solutions.- 4.4 The method of invariant majorants.- 5. Examples.- 5.1 Let x ? ?n, a ? ?.- 5.2 Let us illustrate the algorithm for computing the group admitted by a differential equation by means of the example of a second-order equation.- 5.3 The Korteweg-de Vries equation.- 5.4 Consider the equation of motion of a polytropic gas.- 1: Motions in Riemannian Spaces.- 6. The general group of motions.- 6.1 Local Riemannian manifolds.- 6.2 Arbitrary motions in Vn.- 6.3 The defect of a group of motions in Vn.- 6.4 Invariant family of spaces.- 7. Examples of motions.- 7.1 Isometries.- 7.2 Conformal motions.- 7.3 Motions with ? = 2.- 7.4 Nonconformal motions with ? = 1.- 7.5 Motions with given invariants.- 8. Riemannian spaces with nontrivial conformal group.- 8.1 Conformally related spaces.- 8.2 Spaces of constant curvature.- 8.3 Conformally-flat spaces.- 8.4 Spaces with definite metric.- 8.5 Lorentzian spaces.- 9. Group analysis of Einstein's equations.- 9.1 Harmonic coordinates.- 9.2 The group admitted by Einstein's equations.- 9.3 The Lie-Vessiot decomposition.- 9.4 Exact solutions.- 10. Conformally-invariant equations of second order.- 10.1 Preliminaries.- 10.2 Linear equations in Sn.- 10.3 Semilinear equations in Sn.- 10.4 Equations admitting an isometry group of maximal order.- 10.5 The wave equation in Lorentzian spaces.- 2: A Group-Theoretical Approach to the Huygens Principle.- 11. General considerations and some history of the problem.- 11.1 Hadamard's problem.- 11.2 Hadamard's criterion.- 11.3 The Mathisson-Asgeirsson Theorem.- 11.4 The necessary conditions of Gunther and McLenaghan.- 11.5 The Lagnese-Stellmacher transformation.- 11.6 The present state of the art and generalizations of Hadamard's problem.- 12. The wave equation in V4.- 12.1 Computation of the geodesic distance in a plane-wave metric.- 12.2 Conformal invariance and the Huygens principle.- 12.3 The solution of the Cauchy problem.- 12.4 The case of a trivial conformal group.- 13. The Huygens principle in Vn+1.- 13.1 Preliminary analysis of the solution.- 13.2 The Fourier transform of the Bessel function J0(a ? ).- 13.3 The descent method. Representation of solution for arbitrary n.- 13.4 Summary of the Huygens principle.- 13.5 Failure of the connection between Huygens' principle and conformal invariance.- II: Tangent Transformations.- 3: Introduction to the Theory of Lie-Backlund Groups.- 14. Heuristic considerations.- 14.1 Contact transformations.- 14.2 Finite-order tangent transformations.- 14.3 Bianchi-Lie transformation.- 14.4 Backlund transformations. Examples.- 14.5 The concept of infinite-order tangent transformation.- 15. Formal groups.- 15.1 Lie's equation for formal one-parameter groups.- 15.2 Invariants and invariant manifolds.- 16. One-parameter groups of Lie-Backlund transformations.- 16.1 Definition and the infinitesimal criterion.- 16.2 Lie-Backlund operators. Canonical operators.- 16.3 Examples.- 17. Invariant differential manifolds.- 17.1 A criterion of invariance.- 17.2 Examples of solutions of the defining equation.- 17.3 Ordinary differential equations.- 17.4 The isomorphism theorem.- 17.5 Linearization by means of Lie-Backlund transformations.- 4: Equations with Infinite Lie-Backlund Groups.- 18. Typical examples.- 18.1 The heat equation.- 18.2 The Korteweg-de Vries equation.- 18.3 A fifth-order equation.- 18.4 The wave equation