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Laplace's Equation--Toroidal Coordinates


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.
(1)

Attempt separation of variables by plugging in the trial solution

 f(u,v,phi)=sqrt(coshu-cosv)U(u)V(v)Psi(psi),
(2)

then divide the result by csch^2u(coshu-cosv)^(5/2) U(u)V(v)Phi(phi) to obtain

 1/4sinh^2u+coshusinhu(U^'(u))/(U(u))+sinh^2u(U^('')(u))/(U(u)) 
 +sinh^2u(V^('')(v))/(V(v))+(Phi^('')(phi))/(Phi(phi))=0.
(3)

The function Phi(phi) then separates with

 (Phi^('')(phi))/(Phi(phi))=-m^2,
(4)

giving solution

 Psi(psi)=sin; cos(mpsi)=sum_(k=1)^infty[A_ksin(mpsi)+B_kcos(mpsi)].
(5)

Plugging Psi(psi) back in and dividing by sinh^2u gives

 cothu(U^'(u))/(U(u))+(U^('')(u))/(U(u))-(m^2)/(sinh^2u)+1/4+(V^('')(v))/(V(v))=0.
(6)

The function V(v) then separates with

 (V^('')(v))/(V(v))=-n^2,
(7)

giving solution

 V(v)=sin; cos(nv)=sum_(k=1)^infty[C_ksin(nv)+D_kcos(nv)].
(8)

Plugging V(v) back in and multiplying by V(v) gives

 U^('')(u)+cothuU^'(u)-[(m^2)/(sinh^2u)+(n^2-1/4)]U(u)=0,
(9)

which can also be written

 1/(sinhu)d/(du)(sinhu(dU)/(du))-[(m^2)/(sinh^2u)+(n^2-1/4)]U=0
(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 U(u) are known as toroidal functions.


See also

Laplace's Equation, Laplacian, Toroidal Coordinates, Toroidal Function

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References

Arfken, G. "Toroidal Coordinates (xi,eta,phi)." §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.

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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