11 coordinate system for separation of variables

In summary: 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.Rotational5. 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),
  • #1
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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?
 
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  • #2
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
 
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  • #3
lurflurf said:
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?
 
  • #4
See Morse and Feshback, there are others.
 
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  • #5
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.
 

Related to 11 coordinate system for separation of variables

What is an 11 coordinate system for separation of variables?

An 11 coordinate system for separation of variables is a mathematical tool used in scientific research to solve complex equations by breaking them down into simpler parts. It involves using 11 independent variables to separate a multi-dimensional function into a product of one-dimensional functions.

How does the 11 coordinate system work?

The 11 coordinate system works by transforming a multi-dimensional function into a product of one-dimensional functions, which can then be solved separately. This is achieved by choosing 11 independent variables and using them to rewrite the original function in a way that allows for separation of variables.

What are the advantages of using an 11 coordinate system for separation of variables?

The main advantage of using an 11 coordinate system for separation of variables is that it simplifies complex equations, making them easier to solve. It also allows for the identification of patterns and relationships between the variables, which can provide valuable insights into the behavior of the system being studied.

How is the 11 coordinate system different from other coordinate systems?

The 11 coordinate system differs from other coordinate systems in that it uses 11 independent variables instead of the traditional 2 or 3. This allows for a more detailed and comprehensive analysis of complex systems, making it a powerful tool for scientific research.

What are some real-world applications of the 11 coordinate system for separation of variables?

The 11 coordinate system for separation of variables has various applications in fields such as physics, engineering, and economics. It can be used to model and analyze complex systems, such as fluid dynamics, heat transfer, and population dynamics. It is also commonly used in quantum mechanics and statistical mechanics to solve equations and study the behavior of particles and systems.

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