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Lie theory and separation of variables. 10. Nonorthogonal R-separable solutions of the wave equation [partial-derivative]ttpsi=Delta2psi
We classify and discuss the possible nonorthogonal coordinate systems which lead to R-separable solutions of the wave equation. Each system is associated with a pair of commuting operators in the symm...

Lie theory and separation of variables. 9. Orthogonal R-separable coordinate systems for the wave equation psitt−Delta2psi=0

J. Math. Phys. 17, 331 (1976); doi:10.1063/1.522900

Issue Date: March 1976

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E. G. Kalnins
Centre de Recherches Mathématiques, Université de Montréal, Montréal 101, Québec, Canada

W. Miller, Jr.
School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
A list of orthogonal coordinate systems which permit R-separation of the wave equation psitt−Delta2psi=0 is presented. All such coordinate systems whose coordinate curves are cyclides or their degenerate forms are given. In each case the coordinates and separation equations are computed. The two basis operators associated with each coordinate system are also presented as symmetric second order operators in the enveloping algebra of the conformal group O(3,2). Journal of Mathematical Physics is copyrighted by The American Institute of Physics.
Permalink: http://link.aip.org/link/?JMAPAQ/17/331/1
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PACS

  • 02.20.Qs
    Mathematical methods in physics Group theory General properties, structure, and representation of Lie groups
  • YEAR: 1976

PUBLICATION DATA

ISSN:
0022-2488 (print)   1089-7658 (online)
Publisher:
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REFERENCES (11)
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  1. E. G. Kalnins and W. Miller Jr., J. Math. Phys. 16, xxxx (1975).
  2. P. Moon and D. E. Spencer, Field Theory Handbook (Springer-Verlag, Berlin, 1961).
  3. M. Bócher, Ueber die Reihenentwickelungen der Potentialtheorie (Teubner, Leipzig, 1894).
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  5. J. L. Coolidge, A Treatise on the Circle and the Sphere (Chelsea, New York, 1971).
  6. T. J. I'A. Bromwich, Quadratic Forms and Their Classification by Means of Invariant Factors (Cambridge U.P., Cambridge, 1906).
  7. E. L. Ince, Ordinary Differential Equations (Dover, New York, 1956).
  8. E. G. Kalnins, “On the separation of variables for the Laplace equation [open triangle]psi+K2psi = 0 in two and three dimensional Minkowski space,” SIAM J. Math. Anal. 6, 340 (1975).
  9. E. G. Kalnins and W. Miller Jr., J. Math. Phys. 15, 263 (1974).
  10. N. W. Macfadyen and P. Winternitz, J. Math. Phys. 12, 281 (1971).
  11. L. P. Eisenhart, Ann. Math. 35, 284 (1934).

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