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…mehr
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.
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.
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.
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