In toroidal coordinates, Laplace's
equation becomes
![cschv(cosu-coshv)^3[partial/(partialu)((sinhv)/(coshv-cosu)partial/(partialu))+partial/(partialv)((sinhv)/(coshv-cosu)partial/(partialv))+partial/(partialphi)((cschv)/(coshv-cosu)partial/(partialphi))]f=0.](/images/equations/LaplacesEquationToroidalCoordinates/NumberedEquation1.svg) |
(1)
|
Attempt separation of variables by plugging
in the trial solution
 |
(2)
|
then divide the result by
to obtain
 |
(3)
|
The function
then separates with
 |
(4)
|
giving solution
![Psi(psi)=sin; cos(mpsi)=sum_(k=1)^infty[A_ksin(mpsi)+B_kcos(mpsi)].](/images/equations/LaplacesEquationToroidalCoordinates/NumberedEquation5.svg) |
(5)
|
Plugging
back in and dividing by
gives
 |
(6)
|
The function
then separates with
 |
(7)
|
giving solution
![V(v)=sin; cos(nv)=sum_(k=1)^infty[C_ksin(nv)+D_kcos(nv)].](/images/equations/LaplacesEquationToroidalCoordinates/NumberedEquation8.svg) |
(8)
|
Plugging
back in and multiplying by
gives
![U^('')(u)+cothuU^'(u)-[(m^2)/(sinh^2u)+(n^2-1/4)]U(u)=0,](/images/equations/LaplacesEquationToroidalCoordinates/NumberedEquation9.svg) |
(9)
|
which can also be written
![1/(sinhu)d/(du)(sinhu(dU)/(du))-[(m^2)/(sinh^2u)+(n^2-1/4)]U=0](/images/equations/LaplacesEquationToroidalCoordinates/NumberedEquation10.svg) |
(10)
|
(Arfken 1970, pp. 114-115). Laplace's equation is partially separable, although the Helmholtz
differential equation is not.
Solutions to the differential equation for
are known as toroidal
functions.
See also
Laplace's Equation,
Laplacian,
Toroidal Coordinates,
Toroidal
Function
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References
Arfken, G. "Toroidal Coordinates
." §2.13 in Mathematical
Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 112-115,
1970.Byerly, W. E. An
Elementary Treatise on Fourier's Series, and Spherical, Cylindrical, and Ellipsoidal
Harmonics, with Applications to Problems in Mathematical Physics. New York:
Dover, pp. 264-266, 1959.Morse, P. M. and Feshbach, H. Methods
of Theoretical Physics, Part I. New York: McGraw-Hill, p. 666, 1953.Referenced
on Wolfram|Alpha
Laplace's Equation--Toroidal
Coordinates
Cite this as:
Weisstein, Eric W. "Laplace's Equation--Toroidal Coordinates." From MathWorld--A Wolfram Web Resource.
https://mathworld.wolfram.com/LaplacesEquationToroidalCoordinates.html
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