𝑅-separation of variables for the four-dimensional flat space Laplace and Hamilton-Jacobi equations

Kalnins, E. G.; Miller, Willard

doi:10.1090/S0002-9947-1978-0506618-2

$R$-separation of variables for the four-dimensional flat space Laplace and Hamilton-Jacobi equations
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Trans. Amer. Math. Soc. 244 (1978), 241-261 Request permission

Abstract:

All A-separable orthogonal coordinate systems for the complex equations $\Sigma _{i = 1}^4 {{\partial _{ii}}\Psi = 0}$ and $\Sigma _{i = 1}^4 {{{({\partial _i}W)}^2} = 0}$ are classified and it is shown that these equations separate in exactly the same systems.

References

E. G. Kalnins and W. Miller Jr., Lie theory and the wave equation in space-time. I. The Lorentz group, J. Mathematical Phys. 18 (1977), no. 1, 1–16. MR 507308, DOI 10.1063/1.523130
E. G. Kalnins and W. Miller Jr., Lie theory and the wave equation in space-time. I. The Lorentz group, J. Mathematical Phys. 18 (1977), no. 1, 1–16. MR 507308, DOI 10.1063/1.523130

Orthogonal separable coordinates for the Laplace-Beltrami operator on the complex four-sphere

Lie theory and the wave equation in space time

R-separable solutions of the wave equation

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E. G. Kalnins and W. Miller Jr., Lie theory and the wave equation in space-time. I. The Lorentz group, J. Mathematical Phys. 18 (1977), no. 1, 1–16. MR 507308, DOI 10.1063/1.523130

Die Reihentwickelungen der Potentialtheorie

Luther Pfahler Eisenhart, Stackel systems in conformal Euclidean space, Ann. of Math. (2) 36 (1935), no. 1, 57–70. MR 1503208, DOI 10.2307/1968664
C. P. Boyer, E. G. Kalnins, and Willard Miller Jr., $R$-separable coordinates for three-dimensional complex Riemannian spaces, Trans. Amer. Math. Soc. 242 (1978), 355–376. MR 496814, DOI 10.1090/S0002-9947-1978-0496814-5
Willard Miller Jr., Symmetry and separation of variables, Encyclopedia of Mathematics and its Applications, Vol. 4, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1977. With a foreword by Richard Askey. MR 0460751
Parry Moon and Domina Eberle Spencer, Theorems on separability in Riemannian $n$-space, Proc. Amer. Math. Soc. 3 (1952), 635–642. MR 49439, DOI 10.1090/S0002-9939-1952-0049439-7

Über die integration der Hamilton-Jacobischen differentialgleichung mittels separation der variabelen

Luther Pfahler Eisenhart, Separable systems of Stackel, Ann. of Math. (2) 35 (1934), no. 2, 284–305. MR 1503163, DOI 10.2307/1968433
Parry Moon and Domina Eberle Spencer, Separability conditions for the Laplace and Helmholtz equations, J. Franklin Inst. 253 (1952), 585–600. MR 49437, DOI 10.1016/0016-0032(52)90682-0
Luther Pfahler Eisenhart, Riemannian Geometry, Princeton University Press, Princeton, N. J., 1949. 2d printing. MR 0035081