Separations of Variables and Analytic Contractions on Two-Dimensional Hyperboloids
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eparations of Variables and Analytic Contractions on Two-Dimensional Hyperboloids1 G. S. Pogosyana, b, *, ** and A. Yakhnob, *** a
Yerevan State University, Yerevan, Armenia and Joint Institute for Nuclear Research, Dubna, Russia de Matemáticas, CUCEI, Universidad de Guadalajara, Guadalajara, Jalisco, México *e-mail: [email protected] **e-mail: [email protected] ***e-mail: [email protected]
bDepartamento
Abstract—In this review we present recent results in the field of analytical contraction of Lie algebra in twodimensional hyperbolic space. A complete geometric description for all possible orthogonal and nonorthogonal (related to the first order symmetries) systems of coordinates, which allow separation of variables of twodimensional Laplace–Beltrami or Helmholtz equation on the two-sheeted (upper sheet) H 2 and the onesheeted H 2 hyperboloids is given. The limiting transition between non subgroup (mostly parametric) and subgroup systems is conducted. The analytic contractions between various systems of coordinates in two hyperbolic spaces and Euclidean E2 and Minkowski E11, spaces are presented. DOI: 10.1134/S1063779619020035
CONTENTS 1. INTRODUCTION 2. SEPARATION OF VARIABLES ON TWO DIMENSIONAL HYPERBOLOIDS 2.1. First-Order Symmetries and Separation of Variables 2.1.1. Classification of first-order symmetries 2.1.2. Subgroup coordinate systems 2.2. Classification of Second-Order Symmetries 2.3. Second-Order Symmetries and Non-Subgroup Coordinates on Two-Dimensional Hyperboloids 2.4. Semi-Hyperbolic System of Coordinates (SH) 2.5. Semi-Circular-Parabolic System of Coordinates (SCP) 2.6. Elliptic-Parabolic System of Coordinates (EP) 2.7. Hyperbolic-Parabolic System of Coordinates (HP) 2.8. Elliptic System of Coordinates (E) 2.9. Rotated Elliptic System of Coordinates (E )
2.10. Hyperbolic System of Coordinates (H) 2.11. Rotated Hyperbolic System of Coordinates (H ) 3. CONTRACTIONS ON TWO-SHEETED HYPERBOLOID 3.1. Contraction of Lie Algebra so(2; 1) to e(2) 3.2. Transition from the Pseudo-Spherical System of Coordinates Into Polar 3.3. Transition from the Equidistant System of Coordinates Into Cartesian 3.4. Transition from Horocyclic System of Coordinates Into Cartesian 3.5. Contractions of Nonorthogonal Systems 3.6 Contractions in Elliptic System of Coordinates 3.6.1. Transition from elliptic system of coordinates into elliptic 3.6.2. Transition from elliptic system of coordinates into polar 3.6.3. Transition from elliptic system of coordinates into Cartesian 3.6.4. Transition from rotated elliptic system of coordinates into parabolic 3.7. Transition from Hyperbolic System of Coordinates Into Cartesian
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POGOSYAN, YAKHNO
3.8. Contractions of Semi-Hyperbolic System of Coordinates 3.8.1. Transition from semi-hyperbolic into Cartesian 3.8.2. Transition from semi-hyperbolic system of coordinate into parabolic 3.9. Contractions of Elliptic-Parab
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