Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

Separation of Laplace's equation


Authors: N. Levinson, B. Bogert and R. M. Redheffer
Journal: Quart. Appl. Math. 7 (1949), 241-262
MSC: Primary 36.0X
DOI: https://doi.org/10.1090/qam/32091
MathSciNet review: 32091
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The following results are established in this paper:

(I)$ ^{**}$ For the Laplace equation $ \Delta \theta = 0$ in curvilinear co-ordinates $ \left( {u,v,w} \right)$ in Euclidean space to be directly separableinto two equations, one for $ S$ and one for $ Z$, when the solution is $ \theta = R\left( {u,v,w} \right)S\left( {u,v} \right)Z\left( w \right)$ with fixed $ R$, it is necessary and sufficient that the surfaces $ {w}$ = constant (1) be orthogonal to the surfaces $ {u}$ = constant, $ {v}$ = constant and (2) be parallel planes, planes with a common axis, concentric spheres, spheres tangent at a common point, or one of the two sets of spheres generated by the co-ordinate lines when bicircular co-ordinates are rotated about the line joining the poles or about its perpendicular bisector.

(II) We have $ R = 1$ always and only in the first three cases, namely, when the surfaces $ {w}$ = constant are parallel planes, planes with a common axis, or concentric spheres.

(III) In these three cases, but only these, the wave equation separates in the sense $ RSZ$, and hence, for the wave equation, $ R = 1$ automatically.

(IV) For further separation of the equation found above for $ S$, when $ S = X\left( u \right)Y\left( v \right)$ so that the solution is now $ RXYZ$, it is necessary and sufficient that the co-ordinates be toroidal, or such that the wave equation so separates, or any inversions of these.

(V) The co-ordinates where the wave equation so separates, that is, admits solutions $ RX\left( u \right)Y\left( v \right)Z\left( w \right)$, are only the well-known cases where this happens with $ R = 1$, namely, degenerate ellipsoidal or paraboloidal co-ordinates (but see Sec. 8.2).

(VI). In these cases, but only these, $ R = 1$ for the Laplace equation too.

(VII) Co-ordinates for $ RSZ$ or $ RXYZ$ separation of the Laplace equation have the group property under inversion.

(VIII) In all cases $ R$ can be found by inspection of the linear element.


References [Enhancements On Off] (What's this?)

  • [1] W. Blaschke, Eine Verallgemeinerung der Theorie der konfokalen F$ ^{2}$, Math. Zeit. 27, 653-668 (1928). MR 1544932
  • [2] L. P. Eisenhart, Riemannian geometry, Princeton, 1926.
  • [3] L. P. Eisenhart, Separable systems of Stäckel, Ann. Math. 35, 284-305 (1934). MR 1503163
  • [4] J. Weinacht, Über die bedingt-periodischen Bewegung eines Massenpunktes, Math. Ann. 91, 279-299 (1922). MR 1512194
  • [5] P. Stäckel, Habilitationsschrift, Halle, 1891; also Über die Bewegung eines Punktes in einer n-fachen Mannigfaltigkeit, Math. Ann. 42, 537-563 (1893). MR 1510794
  • [6] A. R. Forsyth, Lectures on the differential geometry of curves and surfaces, Cambridge Univ. Press, 1912.
  • [7] M. Bocher, Über die Reihenentwickelungen der Potentialtheorie, Leipzig, 1894 (Dissertation).
  • [8] L. P. Eisenhart, Stäckel systems in conformal Euclidean space, Ann. Math. 36, 57-70 (1935). MR 1503208
  • [9] E. Haskins, Stäckel systems, M. I. T. Course XVIII Thesis, 1936.
  • [10] C. W. Murchison, Jr., Separable Stäckel systems, M. I. T. Course XVIII Thesis, 1937.
  • [11] H. Feshbach, Jr., and P. M. Morse, Methods of theoretical physics, M. I. T. Notes, Chap. I.
  • [12] H. P. Robertson, Bermerkung über separierbare Systeme in der Wellenmechanik, Math. Ann. 98, 749-752 (1928). MR 1512435
  • [13] W. C. Graustein, Differential geometry, The Macmillan Co., New York, 1935.
  • [14] P. Franklin, Methods of advanced calculus, McGraw-Hill, New York, 1944. MR 0010896
  • [15] J. Liouville, Note au sujet de l'article précédent, J. de Math. 12, 265 (1847).

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 36.0X

Retrieve articles in all journals with MSC: 36.0X


Additional Information

DOI: https://doi.org/10.1090/qam/32091
Article copyright: © Copyright 1949 American Mathematical Society

American Mathematical Society