A 11 coordinate system for separation of variables

[Trying2Learn]

TL;DR Summary

Why are there 11?

Good Morning

I have a very vague memory of having read (about 40 years ago) that there are only 11 coordinate systems in which the field equations of physics can be separated.

I can no longer be sure if my memory has failed me. But this issue has been in my head for all these years. (Gotta do something during quarantine; and a good luck to all of you while I am at it.)

Can someone tell me if this is true?

If it is true, can you list the names of the coordinate systems (I can look up, on my own, what they look like).

And, more important... why 11?

 

[lurflurf]

You are probably thinking of the 11 coordinate systems of Eisenhart popularized by Moon and Spencer. The result is not as profound as you recall. The answer changes depending won the coordinates and equations you are interested in. If one is interested in coordinates of degree 1 and 2 in 3 dimentions and equations related to Helmholtz equation there are 11 sets. One might also consider coordinates of other degree in particular 4. The Laplace equation is often separable when other equations are not for example bispherical coordinates. More complicated equations may not be separable when Helmholtz equation is.

some references

EISENHART, L. P.: Separable systems of STACKEL. Ann. Math. 35, 284 (1934). -
Stackel systems in conformal euclidean space. Ann. Math. 36, 57 (1935).
https://en.wikipedia.org/wiki/Quadric
https://mathworld.wolfram.com/OrthogonalCoordinateSystem.html
https://mathworld.wolfram.com/SeparationofVariables.html
https://mathworld.wolfram.com/BisphericalCoordinates.html

some Moon and Spencer References taken from
https://mathworld.wolfram.com/SeparationofVariables.html
Moon, P. and Spencer, D. E. "Separability Conditions for the Laplace and Helmholtz Equations." J. Franklin Inst. 253, 585-600, 1952.
Moon, P. and Spencer, D. E. "Theorems on Separability in Riemannian n-space." Proc. Amer. Math. Soc. 3, 635-642, 1952.
Moon, P. and Spencer, D. E. "Recent Investigations of the Separation of Laplace's Equation." Proc. Amer. Math. Soc. 4, 302-307, 1953.
Moon, P. and Spencer, D. E. "Separability in a Class of Coordinate Systems." J. Franklin Inst. 254, 227-242, 1952.
Moon, P. and Spencer, D. E. Field Theory for Engineers. Princeton, NJ: Van Nostrand, 1961.
Moon, P. and Spencer, D. E. "Eleven Coordinate Systems." §1 in Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 1-48, 1988.

some lists taken from
Moon, P. and Spencer, D. E. "Eleven Coordinate Systems." §1 in Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 1-48, 1988.

The eleven coordinate systems [3J, formed from first and second degree surfaces,
are as follows:
Cylindrical
1. Rectangular coordinates (x, y,z), Fig. 1.01.
2. Circular-cylinder coordinates (r, "P, z), Fig. 1.02.
3· Elliptic-cylinder coordinates ('Y},"P,z), Fig. 1.03.
4. Parabolic-cylinder coordinates (/-l, '1', z), Fig. 1.04.
Rotational
5. Spherical coordinates (r, O,"P), Fig. 1.05.
6. Prolate spheroidal coordinates ('Y}, 0, "P), Fig. 1.06.
7. Oblate spheroidal coordinates ('Y}, 0, "P), Fig. 1.07-
8. Parabolic coordinates (/-l, '1', "P), Fig. 1.08.
General
9. Conical coordinates (r, 0, A), Fig. 1.09.
10. Ellipsoidal coordinates ('Y}, 0, A), Fig. 1.10.
11. Paraboloidal coordinates (/-l, '1', A), Fig. 1.11.

The partial differential equations considered in this book are as follows:
(1) Laplace's equation
(2) Poissons's equation
(3 ) The diffusion equation
(4) The wave equation
(5) The damped wave equation
(6) Transmission line equation
(7) The vector wave equation

Those are the equations related to Helmholtz equation

 

[Trying2Learn]

You are probably thinking of the 11 coordinate systems of Eisenhart popularized by Moon and Spencer. The result is not as profound as you recall. The answer changes depending won the coordinates and equations you are interested in. If one is interested in coordinates of degree 1 and 2 in 3 dimentions and equations related to Helmholtz equation there are 11 sets. One might also consider coordinates of other degree in particular 4. The Laplace equation is often separable when other equations are not for example bispherical coordinates. More complicated equations may not be separable when Helmholtz equation is.

some references

EISENHART, L. P.: Separable systems of STACKEL. Ann. Math. 35, 284 (1934). -
Stackel systems in conformal euclidean space. Ann. Math. 36, 57 (1935).
https://en.wikipedia.org/wiki/Quadric
https://mathworld.wolfram.com/OrthogonalCoordinateSystem.html
https://mathworld.wolfram.com/SeparationofVariables.html
https://mathworld.wolfram.com/BisphericalCoordinates.html

some Moon and Spencer References taken from
https://mathworld.wolfram.com/SeparationofVariables.html
Moon, P. and Spencer, D. E. "Separability Conditions for the Laplace and Helmholtz Equations." J. Franklin Inst. 253, 585-600, 1952.
Moon, P. and Spencer, D. E. "Theorems on Separability in Riemannian n-space." Proc. Amer. Math. Soc. 3, 635-642, 1952.
Moon, P. and Spencer, D. E. "Recent Investigations of the Separation of Laplace's Equation." Proc. Amer. Math. Soc. 4, 302-307, 1953.
Moon, P. and Spencer, D. E. "Separability in a Class of Coordinate Systems." J. Franklin Inst. 254, 227-242, 1952.
Moon, P. and Spencer, D. E. Field Theory for Engineers. Princeton, NJ: Van Nostrand, 1961.
Moon, P. and Spencer, D. E. "Eleven Coordinate Systems." §1 in Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 1-48, 1988.

some lists taken from
Moon, P. and Spencer, D. E. "Eleven Coordinate Systems." §1 in Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 1-48, 1988.

The eleven coordinate systems [3J, formed from first and second degree surfaces,
are as follows:
Cylindrical
1. Rectangular coordinates (x, y,z), Fig. 1.01.
2. Circular-cylinder coordinates (r, "P, z), Fig. 1.02.
3· Elliptic-cylinder coordinates ('Y},"P,z), Fig. 1.03.
4. Parabolic-cylinder coordinates (/-l, '1', z), Fig. 1.04.
Rotational
5. Spherical coordinates (r, O,"P), Fig. 1.05.
6. Prolate spheroidal coordinates ('Y}, 0, "P), Fig. 1.06.
7. Oblate spheroidal coordinates ('Y}, 0, "P), Fig. 1.07-
8. Parabolic coordinates (/-l, '1', "P), Fig. 1.08.
General
9. Conical coordinates (r, 0, A), Fig. 1.09.
10. Ellipsoidal coordinates ('Y}, 0, A), Fig. 1.10.
11. Paraboloidal coordinates (/-l, '1', A), Fig. 1.11.

The partial differential equations considered in this book are as follows:
(1) Laplace's equation
(2) Poissons's equation
(3 ) The diffusion equation
(4) The wave equation
(5) The damped wave equation
(6) Transmission line equation
(7) The vector wave equation

Those are the equations related to Helmholtz equation

Great! Wonderful. Thank You.But now my ridiculoous questoin... WHY ELEVEN?

What is so special?

 

[Dr Transport]

See Morse and Feshback, there are others.

 

[formodular]

The result, summarized in

Eisenhart - Separable Systems in Euclidean 3-Space

seems to be proven in

https://www.jstor.org/stable/1968433

and seems to depend on

https://www.jstor.org/stable/2306278

If anyone goes through these and gets a sense of the idea of the theorem, proof and where it comes from, post your thoughts here it would be good to see.