“Helmholtz Differential Equation”的概念、定义、习题解题方法及翻译-数学辞典

单词

Helmholtz Differential Equation

释义

Helmholtz Differential Equation

A Partial Differential Equation which can be written in a Scalar version

(1)

or Vector form,

(2)

where

is the Laplacian. When

, the Helmholtz differential equation reduces to Laplace'sEquation. When

, the equation becomes the space part of the diffusion equation.

The Helmholtz differential equation can be solved by Separation of Variables in only 11 coordinate systems, 10 ofwhich (with the exception of Confocal Paraboloidal Coordinates) are particular cases of the ConfocalEllipsoidal system: Cartesian, ConfocalEllipsoidal, Confocal Paraboloidal,Conical, Cylindrical, EllipticCylindrical, Oblate Spheroidal,Paraboloidal, Parabolic Cylindrical,Prolate Spheroidal, and Spherical Coordinates (Eisenhart 1934). Laplace's Equation (the Helmholtz differential equation with ) is separable in the two additionalBispherical Coordinates and Toroidal Coordinates.

If Helmholtz's equation is separable in a 3-D coordinate system, then Morse and Feshbach (1953, pp. 509-510) show that

(3)

where

. The Laplacian is therefore of the form




	(4)

which simplifies to

(5)

Such a coordinate system obeys the Robertson Condition, which means that the Stäckel Determinant is of the form

(6)

Coordinate System	Variables	Solution Functions
Cartesian		exponential, Circular Functions, Hyperbolic Functions
Circular Cylindrical		Bessel Functions, Exponential Functions,Circular Functions
Conical		Ellipsoidal Harmonics, Power
Ellipsoidal		Ellipsoidal Harmonics
Elliptic Cylindrical		Mathieu Function, Circular Functions
Oblate Spheroidal		Legendre Polynomial, Circular Functions
Parabolic		Bessel Functions,Circular Functions
Parabolic Cylindrical		Parabolic Cylinder Functions, Bessel Functions, Circular Functions
Paraboloidal		Circular Functions
Prolate Spheroidal		Legendre Polynomial, Circular Functions
Spherical		Legendre Polynomial, Power,Circular Functions

See also Laplace's Equation, Poisson's Equation, Separation of Variables, Spherical BesselDifferential Equation

References

Eisenhart, L. P. ``Separable Systems in Euclidean 3-Space.'' Physical Review 45, 427-428, 1934.

Eisenhart, L. P. ``Separable Systems of Stäckel.'' Ann. Math. 35, 284-305, 1934.

Eisenhart, L. P. ``Potentials for Which Schroedinger Equations Are Separable.'' Phys. Rev. 74, 87-89, 1948.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 125-126 and 509-510, 1953.

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