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 Günther 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-Bäcklund Groups.- § 14. Heuristic considerations.- 14.1 Contact transformations.- 14.2 Finite-order tangent transformations.- 14.3 Bianchi-Lie transformation.- 14.4 Bäcklund 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-Bäcklund transformations.- 16.1 Definition and the infinitesimal criterion.- 16.2 Lie-Bäcklund 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-Bäcklund transformations.- 4: Equations with Infinite Lie-Bäcklund 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.- § 19. Evolution equations.- 19.1 The algebra AF.- 19.2 The Faà de Bruno formula.- 19.3 The algebra LF.- 19.4 Differential substitutions.- 19.5 Equivalence transformations defined by ordinary differential equations.- § 20. Analysis of second- and third-order evolution equations.- 20.1 m = 2.- 20.2 m = 3.- 20.3 Two systems of nonlinear equations.- § 21. The equation F(x,y,z,p,q,r,s,t) = 0.- 21.1 Analysis of the general case.- 21.2 Classification of the equations s = F(z).- 21.3 A system of two nonlinear equations.- 5: Conservation Laws.- § 22. Fundamental theorems.- 22.1 The Noether identity.- 22.2 The Noether theorem.- 22.3 Invariance on the extremals.- 22.4 The action of the adjoint algebra.- 22.5 First integrals of evolution equations.- § 23. Examples.- 23.1 Motion in de Sitter space.- 23.2 The equation utt + ?2u = 0.- 23.3 The non-steady-state transonic gas flow.- 23.4 Short waves.- § 24. The Lorentz group.- 24.1 Conservation laws in relativistic mechanics.- 24.2 A nonlinear wave equation.- 24.3 Dirac equation.- § 25. The Galilean group.- 25.1 Motion of a particle.- 25.2 Perfect gas.- 25.3 Incompressible fluid.- 25.4 Shallow-water flow.- 25.5 A basis of conservation laws for the K-dV equation.- References.
