Transformation Groups Applied to Mathematical Physics

ISBN-13: 9781402003394 / Angielski / Miękka / 2001 / 394 str.

Nail H. Ibragimov; N. H. Ibragimov

Opis

Approach your problems from the right It isn't that they can't see the solution. end and begin with the answers. Then It is that they can't see the problem. one day, perhaps you will find the final question. G.K. Chesterton. The Scandal of Father Brown 'The Point of a Pin'. 'The Hermit Clad in Crane Feathers' in R.van Gulik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in - gional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in pack ing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "completely integrable systems," "chaos, synergetics and large-scale order," which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics."

Spis treści

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.