Separation of variables and superintegrability : the symmetry of solvable systems /

Separation of variables methods for solving partial differential equations are of immense theoretical and practical importance in mathematical physics. They are the most powerful tool known for obtaining explicit solutions of the partial differential equations of mathematical physics. The purpose of...

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Detalles Bibliográficos
Clasificación:Libro Electrónico
Autores principales: Kalnins, E. G. (Autor), Kress, Jonathan M. (Autor), Miller, Willard (Autor)
Formato: Electrónico eBook
Idioma:Inglés
Publicado: Bristol [England] (Temple Circus, Temple Way, Bristol BS1 6HG, UK) : IOP Publishing, [2018]
Colección:IOP (Series). Release 5.
IOP expanding physics.
Temas:
Separation of variables.
Differential equations, Partial > Numerical solutions.
Mathematical physics.
SCIENCE / Physics / Mathematical & Computational.
Acceso en línea:Texto completo
Tabla de Contenidos:
  • 1. Introduction
  • 2. Background and definitions
  • 2.1. Classical mechanics
  • 2.2. Quantum mechanics
  • 2.3. Integrability and superintegrability
  • 3. Separation of variables
  • 3.1. Some approaches to separability
  • 3.2. The Levi-Civita procedure
  • 3.3. Nonorthogonal separation : examples
  • 3.4. Intrinsic characterization of separation
  • 4. Side condition separation
  • 4.1. A generalization of Stäckel form
  • 4.2. Generalized Helmholtz Stäckel form
  • 4.3. Maximal non-regular separation
  • 4.4. Examples of non-regular separability
  • 5. Separation for the real n-sphere
  • 5.1. Jacobi elliptic coordinates
  • 5.2. Killing vectors and tensors
  • 6. Separation for real Euclidean n-space
  • 6.1. Elliptic coordinates in Euclidean space
  • 6.2. Parabolic coordinates in Euclidean space
  • 6.3. Construction of all separable coordinates
  • 6.4. Comments and references
  • 7. Separation on the hyperboloid
  • 7.1. Branching rules for hyperbolic n-space
  • 7.2. Separation for hyperbolic three-space
  • 8. Conformally flat spaces
  • 8.1. Hyperspherical coordinates
  • 8.2. Separable coordinates : analytic theory
  • 8.3. Separable coordinates : algebraic theory
  • 8.4. Comments and references
  • 9. Time-dependent equations
  • 9.1. Case (i) : time as ignorable variable
  • 9.2. Case (ii) : time-dependent Hamiltonians
  • 9.3. Coordinates on spheres and Euclidean spaces
  • 9.4. Examples
  • 10. Generalized Lie symmetries
  • 11. Differential Stäckel form
  • 11.1. Separation of Laplace equations
  • 12. Functional separation
  • 12.1. A forced wave equation
  • 12.2. Pseudo-Riemannian spaces
  • 13. Vector equations
  • 13.1. Dirac-type equations
  • 14. Links with r-matrix theory
  • 14.1. Complex constant curvature spaces
  • 14.2. Generic ellipsoidal coordinates
  • 14.3. Cyclidic coordinates
  • 15. Multiseparability
  • 15.1. 2D superintegrable systems
  • 15.2. Canonical equations
  • 15.3. 3D superintegrable systems
  • 15.4. Conclusions and extensions.

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