---

＜電子ブック＞
Nonlinear Symmetries and Nonlinear Equations / by G. Gaeta
(Mathematics and Its Applications ; 299)

版	1st ed. 1994.
出版者	(Dordrecht : Springer Netherlands : Imprint: Springer)
出版年	1994
本文言語	英語
大きさ	XIX, 258 p : online resource
著者標目	*Gaeta, G author SpringerLink (Online service)
件　名	LCSH:Differential equations LCSH:Mathematical physics FREE:Differential Equations FREE:Theoretical, Mathematical and Computational Physics
一般注記	I - Geometric setting -- a): Equations and functions as geometrical objects -- b): Symmetry -- References -- II - Symmetries and their use -- 1. Symmetry of a given equation -- 2. Linear and C-linearizable equations -- 3. Equations with a given symmetry -- 4. Canonical coordinates -- 5. Symmetry and reduction of algebraic equations -- 6. Symmetry and reduction of ODEs -- 7. Symmetry and symmetric solutions of PDEs -- 8. Conditional symmetries -- 9. Conditional symmetries and boundary conditions -- References -- III - Examples -- 1. Symmetry of algebraic equations -- 2. Symmetry of ODEs (one-soliton KdV) -- 3. Symmetry of evolution PDEs (the heat equation) -- 4. Table of prolongations for ODEs -- 5. Table of prolongations for PDEs -- IV - Evolution equations -- a): Evolution equations - general features -- b): Dynamical systems (ODEs) -- c): Periodic solutions -- d): Evolution PDEs -- References -- V - Variational problems -- 1. Variational symmetries and variational problems -- 2. Variational symmetries and conservation laws: Lagrangian mechanics and Noether theorem -- 3. Conserved quantities for higher order variational problems: the general Noether theorem -- 4. Noether theorem and divergence symmetries -- 5. Variational symmetries and reduction of order -- 6. Variational symmetries, conservation laws, and the Noether theorem for infinite dimensional variational problems -- References -- VI - Bifurcation problems -- 1. Bifurcation problems: general setting -- 2. Bifurcation theory and linear symmetry -- 3. Lie-point symmetries and bifurcation -- 4. Symmetries of systems of ODEs depending on a parameter -- 5. Bifurcation points and symmetry algebra -- 6. Extensions -- References -- VII - Gauge theories -- 1. Symmetry breaking in potential problems and gauge theories -- 2. Strata in RN -- 3. Michel’s theorem -- 4.Zero-th order gauge functionals -- 5. Discussion -- 6. First order gauge functionals -- 7. Geometry and stratification of ? -- 8. Stratification of gauge orbit space -- 9. Maximal strata in gauge orbit space -- 10. The equivariant branching lemma -- 11. A reduction lemma for gauge invariant potentials -- 12. Some examples of reduction -- 13. Base space symmetries -- 14. A scenario for pattern formation -- 15. A scenario for phase coexistence -- References -- VIII - Reduction and equivariant branching lemma -- 1. General setting (ODEs) -- 2. The reduction lemma -- 3. The equivariant branching lemma -- 4. General setting (PDEs) -- 5. Gauge symmetries and Lie point vector fields -- 6. Reduction lemma for gauge theories -- 7. Symmetric critical sections of gauge functionals -- 8. Equivariant branching lemma for gauge functionals -- 9. Evolution PDEs -- 10. Symmetries of evolution PDEs -- 11. Reduction lemma for evolution PDEs -- References -- IX - Further developements -- 1. Missing sections -- 2. Non Linear Superposition Principles -- 3. Symmetry and integrability - second order ODEs -- 4. Infinite dimensional (and Kac-Moody) Lie-point symmetry algebras -- 5. Symmetry classification of ODEs -- 6. The Lie determinant -- 7. Systems of linear second order ODEs -- 8. Cohomology and symmetry of differential equations -- 9. Contact symmetries of evolution equations -- 10. Conditional symmetries, and Boussinesq equation -- 11. Lie point symmetries and maps -- References -- X - Equations of Physics -- 1. Fokker-Planck type equations -- 2. Schroedinger equation for atoms and molecules -- 3. Einstein (vacuum) field equations -- 4. Landau-Ginzburg equation -- 5. The ?6 field theory (three dimensional Landau-Ginzburg equation) -- 6. An equation arising in plasma physics -- 7. Navier-Stokes equations -- 8. Yang-Mills equations.-9. Lattice equations and the Toda lattice -- References -- References and bibliography The study of (nonlinear) dift"erential equations was S. Lie's motivation when he created what is now known as Lie groups and Lie algebras; nevertheless, although Lie group and algebra theory flourished and was applied to a number of dift"erent physical situations -up to the point that a lot, if not most, of current fun­ damental elementary particles physics is actually (physical interpretation of) group theory -the application of symmetry methods to dift"erential equations remained a sleeping beauty for many, many years. The main reason for this lies probably in a fact that is quite clear to any beginner in the field. Namely, the formidable comple:rity ofthe (algebraic, not numerical!) computations involved in Lie method. I think this does not account completely for this oblivion: in other fields of Physics very hard analytical computations have been worked through; anyway, one easily understands that systems of dOlens of coupled PDEs do not seem very attractive, nor a very practical computational tool HTTP:URL=https://doi.org/10.1007/978-94-011-1018-1

 類似資料