Separability in a class of coordinate systems

SEPARABILITY IN A CLASS OF COORDINATE SYSTEMS BY PARRY M O O N 1 AND DOMINA EBERLE SPENCER 2 SUMMARY

A study is made of separability conditions for the Helmholtz and Laplace equations. Attention is confined to the most useful case for physical applications; namely, (a) The coordinate system is either cylindrical or it has rotational symmetry, and (b) The potential is independent of the third space variable. Necessary and sufficient conditions are tabulated, by which one can easily determine if a proposed coordinate system will allow solutions by the method of separation of variables. INTRODUCTION

Separation of variables constitutes a powerful tool for the solution of the partial differential equations of mathematical physics (1). 3 Unfortunately, however, m a n y coordinate systems do not allow separation. For instance, Eisenhart (2) has shown that, in euclidean 3-space with potential a function of all three space variables, simple separability of the Schr6dinger equation is possible in only eleven coordinate systems. And of the eleven, three (the general conical, paraboloidal, and ellipsoidal) are so complicated that they have little practical utility. The apparatus of physics and engineering customarily exhibits some kind of s y m m e t r y ; so in dealing with the field problems t h a t arise in these subjects, one will naturally employ either some form of cylindrical coordinate system or an axially symmetric system. Also, in m a n y cases the potential is a function of only two variables instead of three. It seems reasonable that these simplified conditions should lighten the separability restrictions that apply to the general three-dimensional problem (3). T h e present paper shows how separability conditions for the general case are modified when the potential becomes a function of only two variables. We shall confine ourselves to euclidean 3-space and to orthogonal coordinate systems. It is well known that separation is never possible unless the coordinate surfaces constitute a triply orthogonal set, so all skew systems can be discarded. Two types of coordinate systems will be considered:

(a) Cylindrical Systems. The coordinate surfaces consist of a family of parallel planes and two families of cylinders, the generators of ~ Department of Electrical Engineering, Massachusetts Institute of Technology, Cambridge, Mass. 2 Department of Mathematics, University of Connecticut, Storrs, Conn. a The boldface numbers in parentheses refer to the references appended to this paper. 227

228

PARRY MOON AND DOMINA ]~BERLE SPENCER

[J. F. I.

which are orthogonal to the planes. A cylindrical system can be generated by taking any orthogonal map in a plane and translating the plane in a direction perpendicular to itself. (b) Rotational Systems. Rotational or axially symmetric coordinate systems may be generated by rotating an orthogonal map about an axis. The axis of rotation is usually an axis of symmetry for the plane map. The coordinate surfaces consist of two axially symmetric families which intersect orthogonally, plus a set of half-planes which terminate on the axis of rotation. We shall impose the further restriction that the potential is a function of only two variables. In cylindrical coordinates (u 1, u 2, z), the generator of the cylinders will be taken parallel to the z-axis and the potential will be independent of z. In rotational coordinates (u 1, u 2, ~b), the z-axis is taken as the axi~ of rotation and ~b is the angle about this axis. With either type of coordinate system, the potential may be written = ~(ul, u 0 .

It is convenient to distinguish two classes of separability: simple A partial differential equation in n independent variables is simply separable if, when the potential is expressed as the product

separability and R-separability (3).

(~ =

UI(~/,1) • U2(U2)

"'"

Un(•n),

the equation allows separation into n ordinary differential equations. R-separability occurs when the assumption v , ( u , ) , u~(~:).., u"(u") ¢ =

R(u~, u"-,...u")

(R # const.) leads to the separation of the equation into n ordinary differential equations. Note that, according to these definitions, an equation that allows separation with R = const, is simply separable and is not R-separable. Separability conditions will be investigated for the following cases: 1. R-separability of the Helmholtz equation (a) Cylindrical coordinates (b) Rotational coordinates 2. R-separability of the Laplace equation (a) Cylindrical coordinates (b) Rotational coordinates 3. Simple separability of the Helmholtz equation (a) Cylindrical coordinates (b) Rotational coordinates 4. Simple separability of the Laplace equation (a) Cylindrical coordinates (b) Rotational coordinates

Sept., J 9 ~ 2 . ]

~EPARABILITY IN COORDINATE SYSTEMS

229

As stated previously, the separability conditions d e t e r m i n e d in this paper apply only to the special case where the potential is i n d e p e n d e n t of the third space variable. 1. THE COORDINATESYSTEMS

T r i p l y orthogonal sets of surfaces can be obtained in various ways. For instance, the investigator m a y s t u d y the properties of confocal quadrics or he m a y investigate cyclides, as was done by Bdcher (4). One of the most convenient ways of finding new coordinate systems, however, is b y complex-plane transformations, employing either arbit r a r y functional transformations or using the Schwarz-Christoffel method. T h e resulting m a p is then translated or rotated to give a triply orthogonal family of coordinate surfaces. Let z =

(1)

where z = x + i y , w = u + iv, and where ~ is a n y analytic function. 4 The real and i m a g i n a r y parts are separated, giving two equations

= ~ ( u , v).

(2)

These equations represent (in the z-plane) a family of curves u = const., which intersect orthogonally the family v = coast. We now form coordinate systems for each function ~: a c y l i n d r i c a l system, and one or two r o t a t i o n a l systems. T h e cylindrical system (u 1, u -°, u 3) is represented b y the equations,

~,(u 1, u2), i== 1(u1,

(3)

U 3,

The z-axis is always parallel to the generators of the cylinders, and the coordinate surfaces are u 1 = c o n s t . , u 2 = coast., u 8 = coast. The rotational system (uL u ~, ¢) is {i

~-~ ~I(Ul, U2)'COS~//,

(l(U 1, u2) . sin ~b,

(4)

or { i = ~2(ul, U2) "cOs @,

~2(u I, u 2) sin ¢,

(5)

u,).

The former represents rotation about w h a t was originally the y-axis; the latter, about w h a t was originally the x-axis. T h e z-azis of the rotational system is always t a k e n as the axis of rotation. Rotation is possible also about other lines in the z-plane. 4 Equation 1 imposes no restriction on our coordinate systems. 5: need not be confined to elementary functions but may include functions defined by series and by integrals.

230

PARRY MOON AND DOMINA EBERLE SPENCER

[J. F. I.

Since 5= is an analytic function, the C a u c h y - R i e m a n n equations apply :

0~2

0~1

0~2

Ou I - Ou 2,

O~l

Ou,a

Ou ~.

(6)

F r o m these equations are obtained the relations (which will be used later): O"~i - - + (Ou'p

02~i (Ou~)~

0

(i = 1, 2)

(7)

and

( 0~1)2 Ou--5

( __)20~2 + \ Ou ~

.,+(

-- ( 0~1~2

0~1~ 2

-

~

l

= ( 0~2)+2 k Ou t

(0~2 2 \ Ou 2

T h e metric coefficients are evaluated b y means of the familiar equation,

(0.).

-t-

(0y). (0.). ~

+

tOu

(i = 1, 2, 3)

(9)

or fronl

g" =

+ \ Oy

+ \ Oz

(10)

and 1

gel = ~:.

(11)

For the cylindrical system, Eq. 3, the metric coefficients of Eq. 9 assume the simple form, gal=(O~l)

2

-Ou l

And by Eq. 8, gl1 = g2~.

(13)

Theorem I . For a n y given cylindrical coordinate system obtained directly f r o m Eq. 1, g n and g ~ are equal and are the same as the metric coe~cients obtained f o r the plane m a p , Eq. 2, while gaa is always equal to u n i t y .

For the rotational system, Eq. 4, g n = g22 --

tOu

-Jr-

tOu

gaa --

E,~,(u', u2)-]2.

(14)

Sept., ]952.]

SEPARABILITY IN

COORDINATE SYSTEMS

-oo- r

Again gll and g22 are the same as obtained from the original plane system, Eq. 2. Similarly, for Eq. 5,

gl, = g~2 = k ~ u l /

-t-

~u~],

gaa = [-~.,(u l, u-~)-]L

(15)

Theorem I I . For any rotational coordinate system obtained directly from Eq. 1, g,l and g22 are equal and are lhe same as the metric coeficients obtained for the plane map, Eq. 2. The third coegcient gaa is always equal Io the square of one of the original functions, Eq. 2. I t is sometimes convenient to modify a coordinate system obtained directly from the complex transformation, Eq. 1. The variable u can always be replaced, if desired, by a function of u , and v can be replaced by a function of v :

{ u ~ 0(u), v ~ or(v).

These transformations do not affect orthogonality or separability. alter the metric coefficients so t h a t

(16) In general, however, they

In the interests of simplicity, the present paper is limited to coordinate systems obtained directly from Eq. 1 and represented by Eqs. 3, 4, 5. A n y of these systems, however, can be subsequently rewritten by use of transformation (16), and this change will not affect separability. F o r example, take the transformation x

~e

w.

The m a p in the z-plane is represented by {

X ~

eu

cos V~

y

eu

sin v,

=

and the metric coefficients are gn = go_, = e2,*. In the z-plane, u = const, represents a family of concentric circles a b o u t the origin, while v = const, indicates a set of radial lines. Cylindrical and rotational s y s t e m s are obtained by use of Eqs. 3, 4, 5; and gn = o 2 2 in all cases. As an alternative, one m a y find it convenient to make the further transformation, U = In r or COS O,

eu =

1"9 ~1 =

r

v

O, ~2

~ sin O.

This gives the usual circular-cylinder coordinates,

and the spherical coordinates, Eq. 5,

I

I

X = ~1 = r cos 0, y

~2

z

2

r sin O,

x = ~zcos¢ = rsin0cos¢, y = ~-osln ¢ = r s i n O s i n ¢, Z = ~l = r cos O.

But in these coordinates, gn and g-o-oare no longer equal: gll = 1, g-o-o= r.

PARRY MOON AND

232

DOMINA EBERLE SPENCER

[J. F. I.

We shall now consider the necessary and sufficient conditions for the separation of the Helmholtz and Laplace equations. The general conditions have been formulated elsewhere (3), so it is now necessary merely to state the simplified conditions resulting from the use of our particular coordinates. Attention will be confined to the coordinate systems of Eqs. 3, 4, and 5 with gll = g22 and with the further restriction t h a t the potential is independent of the third coordinate:

= ~(~, u'). 2. R-SElPARABILITY OF T H E HELMI'IOLTZ E Q U A T I O N

T h e Helmholtz equation is

v~ + (k)~ = 0, where k is a constant. T h e necessary and sufficient condition (3) for R-separability of the Helmholtz equation is

i~" 1 0 (f__OR) [~ ] •= figi, Oul 'Ou' + -- (k)2 R = O,

(a)

where the functions Q and f~ are defined by ell = ~

S

Q(.',

,~,.

(b)

. .-"),

and

4 ~ -- [,~ f

SQ

,(u')

]. [-R(u', u~,

. . .

u")-] 2.

(c)

Here S is the St~ickel determinant,

s = ¢~(~) ¢'(ui)"¢~"(~i ) I¢.l(u") ¢.2(u)...¢..(u) and Mil is the cofactor of ~,-1. There are n separation constants a~, O/21

"

"

"

0/n.

An outstanding feature of the St~ckel determinant is that all elements of the first row are functions of the first space coordinate u 1, all elements of the second row are functions of u ~, etc. The elements O~i(u0 of the determinant are arbitrary functions of u ~, though a random choice of functions floes not in general lead to an orthogonal coordinate system in euclidean space. For a given coordinate system, characterized by a given set of metric coefficients gii, the functions '~ij(u ~) are determined so that Eq. b is satisfied. This procedure gives for circular-cylinder coordinates, 0 S=

1

- - -r-2

--1

0

1

0

1

0

1

Sept., 1952.]

SEPARABILITY

IN COORDINATE

SYSTEMS

233

and for spherical coordinates, 1 S~

--

1

¢2

0

1

0

0

1 1 sin 2 0 1

In both of these examples, Q (u 1, u 2, u 3) is 1. With some coordinate systems, however, it is impossible to set up a St/ickel d e t e r m i n a n t satisfying Eq. b unless Q is a function of the coordinates.

For the special cases considered in this paper, R-separability is obtained if the a s s u m p t i o n of

vl(ul), ry (u ) * =

R(u', u-')

(17)

leads to the separation of the partial differential equation into two o r d i n a r y differential equations. Also, the Stiickel d e t e r m i n a n t is a function of only two variables:

S =

(I) 11(U 1) (I)21(U 2)

(1D12(ul) [ 1~22(U2) ] '

b u t ~]g contains all three metric coefficients: 4g = ~g~l go.2 g~3.

Justification for these s t a t e m e n t s can be obtained from the previous paper (3). E q u a t i o n (b) now reduces to gll

=

g22

-

SQ

sO.

Mll

M21

T h u s MI~ = M ~ , or q~22(u2) = -

q'12(u') = const.

W i t h o u t loss of generality, we can set this c o n s t a n t equal to unity, so the St~ckel d e t e r m i n a n t becomes

S ~-- I~21[~11 1 11 = (I)ll(UL) "AI-(I)21(U2)*

(18)

Theorem I I I . I n any coordinate system obtained from Eq. 1, with ~ = (u 1, uS), the Stackel determinant is always equal to the sum of a function of u 1 only and a function of u s only, Eq. 18.

For the Helmholtz equation (3) Q = 1 where A is a constant.

and

~1 = (k) 2 + A ,

T h u s from Eqs. b and 18, we obtain

gll = g22 = S = (IDll(U 1) + (I~21(U2).

(19)

PARRY MOON AND DOMINA EBERLE SPENCER

234

[J. F. I.

F r o m (c) and (a),

4gaa = f~(u~).f2(u2). (R) 2, alO(f i~--Ou~

) iOR ~

(20)

-+- A R S = 0.

(21)

Evidently, Eq. 20 is merely a definition of R and imposes no restriction on the coordinate system. T h e conditions for R-separability of the H e l m h o l t z e q u a t i o n are therefore Eqs. 19 and 21. For a cylindrical system, g4~aa -- 1, so Eq. 20 becomes 1 = fl(u')"f2(u2) • (R)'. W i t h o u t loss of generality, therefore, we can set f~=f~=R=

1.

In other words, separability of the H e l m h o l t z equation in cylindrical coordinates is obtainable only if R = 1. Indeed, this conclusion holds also for the more general case (3) where ~o = ~o(u~, u -~, ua):

Theorem I V . R-separability of the Helmholtz equation is never possible for a cylindrical coordinate system. We therefore t u r n to rotational coordinates. Here ~/~aa = ~, where } = }1 for the system of Eq. 4 and } = (2 for the system of Eq. 5. E q u a t i o n 20 becomes =/l(U')

"f2(u2) • (R) 2.

Thus,

O} = f 2 [ d f l OR] ou UdTu (R? + 2 Rfl 371 or

Ou ~

2

20u*

Ou2

2

~ Ou 2

Similarly,

S u b s t i t u t i o n in Eq. 21 gives

-+- (R)2fif-~--~ ( 0 ~

+ ~

-- 2 f~ (dul)2

f2 (du2) 2 J -t- 4 A S = O.

Utilizing Eqs. 7, 14, 19, 20, we obtain

~u

~

fl(du,)~ +

f2du 2 /

f2(du2)2

(~)2

Sept., ~952.]

SEPARABILITY IN COORDINATE SYSTEMS

235

This equation m a y be written (22)

g " - 9.,(u,) + e . ( u q ,

gaa where

[a,(u,) = ( ~ dr, )~

2 d"-f~ + 4 AdPI1, f l (dul) 2 2 d%,

\ f2 du ~

f2 (du~)~

+ 4 Aovol.

(23)

Theorem V. The necessary and su~cient conditions for R-separability of the Helmholtz equation, in orthogonal rotational coordinates in euclidean 3-space with ~ = ~o(u~, u2), are that gu and gu/gaa be expressible as sums of separated functions: g~, = q,,,(u,) + 4,~(u~),

I

gs~

f~,(u ~) q- ft2(u"),

(~9)

(22)

g33

where t~i are defined by Eq. 23. 3. R-SEPARABILITY OF THE LAPLACE EQUATION

Separability conditions for the Laplace equation, V2~ = O,

nlay be obtained ab initio or t h e y m a y be taken as a special case of conditions for the Helmholtz equation with k = O. Corresponding to Theorem IV, we have

Theorem VI. R-separability of the Laplace equation is never possible j'or a cylindrical coordinate system. For Laplace's equation in rotational 22 and 23 a p p l y with A = as. N o t e t h a t with the Laplace equation than with the is not necessarily a constant b u t m a y be

coordinates, k = 0, and Eqs. the restrictions are less severe Helmholtz, since Q in Eq. (b) entirely arbitrary.

Theorem V I I . The necessary and sufficient condition for R-separability of the Laplace equation, in orthogonal rotational coordinates in euclidean 3-space with ~o = ~(u I, u2), is that gu/gaa be expressible as a sum of separated functions: gil _ t11(u') q- ft2(u2), gaa

where f2~ are defined by Eq. 23 and A = a~.

(122)

236

PARRY MOON AND D O M I N A EBERLE SPENCER

[J. F. I.

4. SIMPLE SEPARABILITY OF THE HELMHOLTZ EQUATION

For simple separability, R = 1 and Eq. 20 becomes = fl(u')'f~(u2).

(20a)

Also, Eq. a reduces to [al -- (k)2Q~ S = O. But since S # 0 and Q = 1, Ogl = (k) 2.

(21a)

And since Q = 1, g l l = (I)ll(~/'1) "~ (I)21(~/,2).

With cylindrical coordinates, ~

(19)

= 1, so Eq. 20a is always satisfied

by fl

=

f2 = 1.

(20b)

Equation 21a evaluates the first separation constant; and neither (21a) nor (21b) imposes any restriction on the coordinate system. So the sole condition for simple separability of the Helmholtz equation in cylindrical coordinates is Eq. 19. With rotational coordinates, Eq. 20a still imposes a restriction : ~g--~ = ~ = flf2 (20a) and Eq. 19 also applies. Theorem V I I I . The necessary and su~cient condition for simple separability of the Helmholtz equation, in cylindrical coordinates obtained f r o m Eq. I and with ~ = , ( u ~, u2), is that gl, be expressible as a sum: g l l = (I~ll(U 1) 71-(I)21(~,2).

(19)

Theorem I X . The necessary and su:ficient conditions for simple separability of the Helmholtz equation, in any rotational coordinate system obtained f r o m Eq. 1 and with ~ = ~(u', u0, are given by Eqs. 19 and 20a. 5. SIMPLE SEPARABILITYOF THE LAPLACE EQUATION

For simple separability of the Laplace equation, R = 1 and k = 0. Returning to Eqs. a, b, and c, we find gll :

SQ,

(19a)

gz3 = f l ( U 1) "f2(u2),

(20a)

al = 0.

(21b)

and With cylindrical coordinates, we m a y write for Eq. 20a, f l = f2 = 1.

(20b)

Since Eqs. 19a and 21b impose no restriction on the coordinates, we conclude t h a t the Laplace equation separates in an infinite n u m b e r of cylindrical coordinate systems:

Sept., 1 9 5 2 . ]

237

SEPARABILITY IN COORDINATE SYSTEMS

Theorem X . The Laplace equation is simply separable in every cylindrical coordinate system obtained f r o m Eq. 1 with ~ = ~(u ~, u~). W i t h r o t a t i o n a l s y s t e m s , 4g-~33 = ( a n d the sole r e q u i r e m e n t for s e p a r a b i l i t y is t h a t g33 be a p r o d u c t : 4~

(20a)

= ~ : f,(u,).f~(u~).

Theorem X I . The necessary and sufficient condition for simple separability of the Laplace equation, in any rotational coordinate system obtained from Eq. 1 with ~ = ¢(u 1, u2), is given by Eq. 20a. 6. THE SEPARATION EQUATIONS

If e i t h e r t h e L a p l a c e or t h e H e h n h o l t z e q u a t i o n separates, t h e s e p a r a t e d e q u a t i o n s are a l w a y s (3)

f~du ~ f~ ~£Tu~

+ U ~ ~ a s ~ j = 0,

(24)

j=l

where i = 1, 2, 3 , . . . n , a n d q~,j are the e l e m e n t s of t h e St~ickel determ i n a n t . F o r t h e c o o r d i n a t e s y s t e m s considered in this paper, n = 2 a n d ~2~ = - 1, q522 = + 1 , so t h e s e p a r a t e d e q u a t i o n s are ----

t -~-u~

-t- Ea,':I:',, -

o~2] U1 =

Lf~du'

' d-~u~

+ E°~le~l + a~] u ~

O,

o.

If t h e c o o r d i n a t e s are cylindrical, f~ = f2 = 1 a n d Eq. 25 b e c o m e s d U1 (du,)~ + F ~ ¢ 1 ~ -

~d

U, = o,

(25a)

d~U ~

(du~)~ + F ~ ¢ ~ + ~ d U 2 = 0. 7. B I P O L A R C O O R D I N A T E S

As an example, t a k e the t r a n s f o r m a t i o n , ~

c(ew+ 1) e. . . . 1 where~ = x--iyandw

= u +iv.

Thus,

x - iy = cE(e~c°sv + 1) + i e ~sinv-] (e ~ c o s v 1) + i e u s i n v cE(e °-~- 1) - - i 2 e ~sinv] e-~" -- 2e ~ cos v + 1, The transformation could equally well be written with z instead of its complex conjugate ?.

238

PARRY M O O N AND DOMINA EBERLE SPENCER

[J. F. I.'

or

f l

F r o m E q . 12,

c sinh u cosh u - cos v '

(26)

c sin v cosh u - cos v"

C2

g n = g,s = (cosh u -

cos v) s"

From the foregoing plane-transformation, c y l i n d r i c a l c o o r d i n a t e s y s t e m (u 1, u 2, u 3) : x

c sinh u 1 c o s h u 1 - cos u 2 '

Y

c sin u s c o s h u t - c o s u 2'

we

now

formulate

a

(27)

Z ~ U3. T h e m e t r i c coefficients a r e

C g l l = g2s ---- (cosh u I -

cos uS) 2'

g~3 = 1.

(28)

T h e s e e q u a t i o n s r e p r e s e n t bipolar-cylinder coordinates. T h e s u r f a c e s u ~ = c o n s t , a r e e c c e n t r i c c i r c u l a r c y l i n d e r s a b o u t t h e p o l a r lines (y = 0, x = ± c). T h e s u r f a c e s u s = c o n s t , a r e c i r c u l a r c y l i n d e r s w i t h c e n t e r s on t h e y-axis, w h i l e u 3 = c o n s t , r e p r e s e n t s a f a m i l y of p a r a l l e l planes. A c c o r d i n g t o S e c t i o n 4, t h e H e l m h o l t z e q u a t i o n s e p a r a t e s if a n d o n l y if

gll = (I)ll(?/tl) .3[-(I)21(U2).

(19)

E v i d e n t l y , E q . 28 d o e s n o t s a t i s f y t h i s c o n d i t i o n , so the Helmholtz equation does not separate i n bipolar-cylinder coordinates. T h e L a p l a c e e q u a t i o n , h o w e v e r , s e p a r a t e s in a n y of o u r c y l i n d r i c a l s y s t e m s . T h i s is s i m p l e s e p a r a b i l i t y , so a l = 0 a n d E q . 25a g i v e s t h e s e p a r a t i o n e q u a tions :

(dul)2 f d y 1 d 2U2

a s U I = O, + ~sU' = O.

T h e s o l u t i o n s of these=~ordinary d i f f e r e n t i a l e q u a t i o n s a r e e x p o n e n t i a l s a n d c i r c u l a r f u n c t i o n s , a n d t h e final s o l u t i o n of a D i r i c h l e t or a N e u m a n n p r o b l e m in b i p o l a r - c y l i n d e r c o o r d i n a t e s is o b t a i n e d as a s u m of t e r m s of the form, = U ~ . U s.

Sept., I952.]

SEPARABILITY IN COORDINATE SYSTEMS

239

T u r n i n g n o w to t h e r o t a t i o n a l c o o r d i n a t e s o b t a i n e d from t h e plane m a p , Eq. 26, we h a v e t w o possibilities. R o t a t i o n of t h e m a p a b o u t its y-axis gives toroidal coordinates." I x

c sinh u I cos ¢ cosh u x - cos u 2 '

Y

cos~ ~u ~ cot ~u'

z

cosh u'

c sinh u ~ sin ¢

(29)

c s i n u -~ -

cos u 2'

with (2

gll = g22 = (cosh u' C2

--

cos u2) 2' (30)

sinh 2 u '

gaa = (cosh u' -- cos u2) ~" R o t a t i o n a b o u t t h e x-axis of the plane m a p , Eq. 26, gives bispherical

coordinates: x =

c sin u'-' cos ¢ coshu l-cosu

2'

c sin u °- sin ¢ Y = c o s h u 1 - c o s u 2' g =

c sinh u 1 cosh u' - c o s

(31)

u" '

with C2

g u = g~2 = (cosh u' -- cos u2) "~ and c 2 s i n 2 gt 2

gaa = (cosh u' - cos u2) 2"

(32)

N o w consider s e p a r a b i l i t y in these r o t a t i o n a l coordinates. According to T h e o r e m s V a n d V I I I , t h e H e l m h o l t z e q u a t i o n s e p a r a t e s o n l y if g u is a separable s u m : g u = (I)ll(U 1) -~-

(~21(U2).

Since n e i t h e r toroidal nor bispherical c o o r d i n a t e s s a t i s f y this c o n d i t i o n , the H e l m h o l t z e q u a t i o n does n o t s e p a r a t e (simple or R) in these coordinates. F o r simple s e p a r a b i l i t y of t h e L a p l a c e e q u a t i o n , t h e n e c e s s a r y a n d sufficient c o n d i t i o n ( T h e o r e m X I ) is gaa = fl(u')"fe(u2), which is n o t satisfied. W e conclude t h a t L a p l a c e ' s e q u a t i o n is n o t s i m p l y separable in toroidal or bispherieal coordinates.

240

PARRY MOON AND D O M I N A E B E R L E S P E N C E R

[J. F. I.

T h e remaining possibility is R-separability of the Laplace equation. W i t h toroidal coordinates, gll g33

1 (sinh ul) 2'

which is a simple case of the required form, fh(u') + ft2(u2).

Also,

62

Q = (cosh u 1 - cos u2) 2' 1

R=

~/cosh u 1 - cos u 2 ' 011 = 1, ¢21 = 0, f l = c s i n h u 1, f2 = 1. S u b s t i t u t i o n in Eq. 23 shows t h a t this equation is also satisfied and al = [. Therefore the Laplace equation is R-separable in toroidal coordinates. Similarly in bispherical coordinates, gll g33 which satisfies Eq. 26.

1 (sin u2) 2'

Also, C2

Q -- (cosh u I - cos u2) 2' R=

1

4cosh u 1 -- cos u ; '

¢11 = 0,

¢21 = 1,

f l = 1,

E q u a t i o n 23 is again satisfied and al = R-separable in bispherical coordinates.

f2 = c s i n u

s.

¼, so Laplace's equation is

8. ONE-DIMENSIONAL SOLUTIONS

T h e previous p a r t of the p a p e r has dealt with solutions in which the potential is a function of t w o variables: =

u2).

W e n o w consider one-dimensional s o l u t i o n s , , = ~(ui), of the H e l m h o l t z and Laplace equations. Since in this special case there is no separation of variables, one might suppose t h a t solutions would be possible in all orthogonal coordinate systems. B u t such is far from being the case. T h e H e l m h o l t z equation in 3-space is ,-1 ou-

Sept., 1952.]

SEPARABILITY IN COORDINATE SYSTEMS

241

A s s u m e t h a t a solution exists in terms of a single coordinate, say ui: =

T h e n two of the derivatives of Eq. 33 become zero, and the equation reduces to e.du;

+

~

~

(k)',p = 0

or

-----2 (duo + [ 4g ~

+ (k)2gJJ~° =- 0.

(34)

A one-dimensional solution is possible only if the coefficients in Eq. 34 are constants or functions of ui only. For the second t e r m of Eq. 34, it is necessary t h a t gii

= f~(uO" Fi,

where Fi is a function of the other two variables (not ui). third term, g~i = gjj(uJ).

(35) For the (36)

Theorem X I I . The necessary and su:ficient conditions that the Helmholtz equation have a solution ~ = q(ui) are

f

4g = fj(ui). F~, giJ

L~

g~j(uO.

(35) (36)

Theorem X I I I . The necessary and suficient condition that the Laplace equation have a solution ~ = ~(ui) is

gii

= fi(uO" Fi.

(35)

T h e r e q u i r e m e n t s of T h e o r e m X I I are h a r d l y ever satisfied. Exa m i n a t i o n of the eleven coordinate systems of Eisenhart (2) shows t h a t only in rectangular coordinates are Eqs. 35 and 36 completely satisfied. Beyond this one coordinate system, there exist only a few isolated onedimensional solutions of the Helmholtz e q u a t i o n (for instance, in spherical coordinates ~o = ~o(r) b u t not ~0 -~ ~o(0) or ~o --- ~o(~b)). T h e Laplace equation is more fortunate. In fact, it is easily shown t h a t an infinite n u m b e r of coordinate systems satisfy T h e o r e m X I I I :

Theorem X I V . Laplace's equation allows one-dimensional solutions in every orthogonal cylindrical coordinate system.

242

.Js~ARRY MOON" AND D O M I N A EBERLE SPENCER

[j. JY. i.

TABLE I.--Separability Conditions.

Equation Helmholtz

For coordinate systems obtained from Eq. 1 with ¢ = ~o(u~, u2). Type of Separation Cylindrical Coordinates Rotational Coordinates R

No separation

gl---!1 fh(u ~) -b ft,(u2)* ga3

Laplace

R

No separation

Helmholtz

Simple

gn = ~ n ( u ~) q- ~2~(u2)

Laplace

Simple Always separable "l~ 2 a,f~ • Where ~i(uO = {\ ~laf~ ~ ] fl (duO~ q'- 4Aei,i~.

gt_.11= ~x(ul) --I-f~2(u2)*

g~3

~ g l l = ~ll(UI) -[- q~l(u2) ~/~

fl(u')

~ / ~ = ~ = f l ( u x) .fs(u 2)

REFERENCES

(1) W. E. BYERLY, "Fourier Series," Boston, Ginn & Co., 1893. (2) L. P. EISENrU,ET, "Separable Systems of St~ickel," A n n . Math., Vol. 35, p. 284 (1934). (3) P. MOON AND D. E. SPENCER, "Separability Conditions for the Laplace and Helmholtz Equations," JOUR. FRANKLIN INST., Vol. 253, p. 585 (1952). (4) M. B6CHER, "Reihenentwickelungen der Potentialtheorie," Berlin, B. G. Teubner, 1894.