Rend. Sem. Mat. Univ. Poi. Torino Voi. 50, 4 (1992)

Differential Geometry

S. BenentiW

INERTIA T E N S O R S A N D STÀCKEL S Y S T E M S IN T H E E U C L I D E A N SPACES

A b s t r a c t . The Cartesian components of the Killing tensors belonging to

the fundamental Stàckel systems in the Euclidean affine space E n (elliptic,

parabolic and spherical-elliptic) are computed within the Stàckel-Eisenhart

theory of the orthogonal separation on a Riemannian manifold, by using the

fundamental properties of the elementary symmetric functions and of the

inertia tensors of a massive body.

1. Introduction

A system of orthogonal separable coordinates on a Riemannian manifold (Mn,g) is geometrically characterized by an n-diménsional space S of Killing 2-tensors over M, whose elements are in involution and have n eigenvectors in common (the intrinsic characterization of the orthogonal separation, after the cruciai results of Eisenhart [Eli] [EI2] [EI3], have been treated in [WO] [KM1] [SH] [BKW] [BKM] [BEI]). Let us cali such a space S a Stàckel system.

The aim of the present lecture is to illustrate a method far calculating the Stàckel systems in the affine Euclidean spaces E n .

Separable coordinates in the Euclidean spaces E n have been fully discussed and classified by Kalnins and Miller [KM2] [KA] after proving that

(*)Research supported by GNFM-CNR and by MURST. Lecture delivered at Giornate di Geometria. Differenziale, Dipartimenti di Matematica dell'Università e del Politecnico di Torino, Torino, May 29-30, 1992.

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in these spaces every separable coordinate system has an orthogonal equivalent (this analysis is extended to other spaces of Constant curvature, the spheres S n [KM2] and the hyperboloids H n [KM3], to the conformally Euclidean spaces and other spaces [KMR] [BKM] [BKW]).

In the- Euclidean spaces E n the elliptic coordinates play a fundamental role. In elliptic coordinates a basis of the corresponding Stackel system can be computed by a method suggested by Jacobi himself (see [M01] [M02]), or, in a more general way, by the theorem of Stackel [STI] [ST2] (see [KM2]). However, for a better understanding of the separation in E n , it is of some interest to know the Cartesian components of the elements of the elliptic Stackel systems. These components can be computed by using the transformation rule of the components of a tensor. However, looking at the transformations from elliptic coordinates to Cartesian coordinates, this seems to be an unpracticable way. So that some special procedure is needed.

A solution to this problem can be found in [MW]. We present here a different approach, which is based on the general theory of orthogonal separation and it is suggested by a known fact concerning the inertia tensor of a massive body: the inertia tensor is a symmetric 2-tensor fìeld whose eigendirections span orthogonal distributions which are completely integrable and whose integrai manifolds are confocal quadrics. Indeed, as it is classically known [EU] [BL] [WE], the coordinate surfaces corresponding to a separable system in E n are confocal quadrics (possibly with degeneracy). Thus there is a rather surprising link between these two apparently distant subjects, the separablity of the Hamilton-Jacobi equation and the geometry of massive bodies. This is simply due to the fact that the inertia, tensor is a Killing tensor.

We shall use this fact (Section 3), together with the main properties of the elementary symmetric functions (which are listed in Section 2), as a tool for dealing with the separation in the Euclidean space E n . In the present lecture we will consider only three kinds of Stackel systems, elliptic (Section 4) parabolic (Section 5) and spherical-elliptic (Section 6), which are the building blocks for ali Stackel systems and ali separable dynamical systems in E n . Elliptic and parabolic coordinates are naturally deiìned by the inertia tensor of a body, more precisely by the planar moment of inertia, denoted by X, which is a conformai Killing tensor and plays the role of generator of the whole Stackel system according to a rule concerning a particular kind

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of orthogonal separable coordinates (Section 2). We allow the masses of the single points of the body to be either positive or negative ìvumbers. When the total mass m is not zero, the center of mass is well dehned, and when the principal moments of inertia at the center of mass are distinct numbers, we get the elliptic coordinates. When m = 0, the center of mass is not dehned and it is substituted by a point O and a vector w, such that LQ • w = 0. If the tensor L has ali distinct eigenvalues at O, then we get the parabolic coordinates (the parabolic axis is determined by the pair (0,w)). Finally, the Stackel system corresponding to spherical-elliptic coordinates (also called conical coordinates) can be obtained as the part proportional to the total mass m of the elliptic Stackel system, or as the limit when the total mass increases to infini ty.

2. Elementary symmetric functions and orthogonal separation

Before dealing with the orthogonal separation in the Euclidean spaces we ought to consider a very special kind of separable coordinates on a generic Riemannian manifold (M n , ^ ) , whose properties are strictly related to the properties of the elementary symmetric functions.

Let aa be the elementary symmetric function of order a — 1 , . . . , n of the n variables (ul), that is the sum of ali possible products of a variables with distinct indices. Let ala"

3 be the symmetric function of order a over the same variables, with the exclusion of those of index (z , . . . , j ) , that is

ì--j _ i • • •

0"a — °"a|w ;= . . .= w j=o-

Let us set

0o = °lo = cr'o = 1, °-\ = ^i-i = tfJ?i = 0; (2-1)

*; =

318

where

di = dul '

from which we derive the identities

dicri =

319

By applying to this last identity the partial derivative di, due to (2.3) and (2.8), we get

n

£ ( - 1 ) * ((n - *) (tt'")»-*-! ck + («•")»-* a'_i) = 0-k=0

i.e.

(2.io) D-1)*( t tT"fc4-i = - ^ V ) .

By applying cfy to (2.9), with j 7̂ i, we get

n

(2.11) B-^^r'i^0-k=i

From (2.10) and (2.11) it follows that if we set

/ i\n-a-l

(2-12) a ? = ( - 1 ) - L ^ _ _ (a = 0 > . . . , n - . l ) ,

then the matrix (a^) is the inverse matrix od (crla), with a = 0 , . . . , n — 1:

n - i

a=0

The determinant of the matrix (aa ap to the factors (—l)a^Ur(ut))"1 is the Vandermonde determinant, so that

• j - r n - l / _ - j \ a d-"''-fc^n"'-'>' where

n—1 n

ji(-i)a=(-i)*n(n""i)J- n . ^ V )= ( - i ) ^ - 1 ^ ' - ujf.

a=0 i=l

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Thus

^ det^ > = u-^ry and

(2-14) det(aj) = J~[(ul - uj). j>i

Finally, from (2.4)3 we get the last identity we will use in the following discussion,

(2.15) £ ^ - 1 ( « ' - « ' ' ) = ***-**• a=0

PROPOSITION 2.1. Let (ul) be orthogonal coordinates 011 a Riemannian manifold (Mn,g). Let (Ei) be the corresponding fra,me ofvector helds. If

(2.16) V V u ' ' , ^ 1 1 1 ^ ' = - ^ ( i ^ i ) ,

or equivalenti)?, if thè 2-tensor tìeld L with eigenvalues (ul) and eigenvectors (Ei),

n (2.17) l=£iu

igiiEi®.Éi i=i

is a conformai Killing tensor, then the coordinates (ul) are separable and the corresponding Stackel system Sis genera,ted by the independent Killing tensors (Ka) (a = 0 , 1 , . . . , n — 1) defined by one of the following equivalent formulae

n (2.18) Ka = '£a

iag"Ei®Ei

1 = 1

(2.19) Ka = aag-Ka-1-L, K^ = 0„

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(2.20) Ka = Yji-l)ka^kL

k

k=0

(2.21) K0 = g, Ka = ^tr(Ka-1-L)g-Ka-1'L (a = 1 , . . . ,n - 1 ) .

To prove this proposition we need two lemmas.

LEMMA 2 .1 . Let L be a symmetric tensor which is diagonalized with respect to orthogonal coordinates (V) . Let (QÌ) be the correspoiiding eigenvalues,

n

(2.22) L^Y^Qig^Ei^Ei.

Tiien X is a conformai Killing tensor if and only if there exists a vector field C = ClEi sudi that the following equations are satisfied

(2.23) diQj = (Qi - Qj)di In gii + Cf (i ? j), diQi = C,-.

L is a Killing tensor when C = 0.

Proof. A tensor field L is a conformai Killing tensor if and only if

{EL,Eg} = EcEg

where 1 •• 1 ••

EL = 2L%3'PiPJ>' E9 = 2gl)piP^ EC = C%p*'

Under the assumption (2.22) we get

E E » " [*" d^i - (ft - erì-digii - 933 Ci i j

and this condition is equivalent to equations (2.23). •

PÌP) = °>

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LEMMA 2.2. An orthogonal coordinate system (ul) is sepa.ra.hle if and only ifthe linear differenti al system

(2.24) diQj = (Qi- Qj)dih\gn (i ^ j), #,•£,• = 0 ,

is completely integrable. If (gaj) (a = 0 , . . . , n — 1) is a complete solution, then the tensors

n

(2.25) Ka^^Qai^Ei^Ei,

form a basis of the St achei system S correspondìng to the separable coordinates (ul).

Proof. The separability conditions for an orthogonal coordinate system, that is equations

didi9hh - di In gii djg

hh - ^ In g" dighh = 0 (i ? j) ,

coincide with the integrability conditions of (2.24) (for a discussion on this topic see [BEI]). •

Proof of Proposition 2.1. Due to Lemma 1 (equations (2.23)) the tensor L defìned by (2.17) is a conformai Killing tensor if and only if conditions (2.16) are satished (when QÌ = ul equations (2.23) become 0 = (ul — u3)d{ ìng33 + C% and Ci = 1). Due to equations (2.16) the differential system (2.24) becomes

(2.26) die) = ~ ~ l ( i / i ) , diQi = 0.

Formula (2.5) shows that this system is identically satished by the functions

(2.27) Qai = *Ì ( a > 0 , . . . , 7 Z - l )

on the domain D = {ul ^ u3 ,i ^ j}. Formula (2.14) shows that in this domain det(£az-) ^ 0. So we ha,ve proved that the linear differential system (2.26) is completely integrable and we know that the functions (2.27) form a basis of solutions. Then, due to Lemma 2.2, the coordinates (ul) are separable and the tensors (2.18) provide a basis for the corresponding Stàckel system. Let us consider these tensors as linear operators (the metric tensor g = KQ is the

http://sepa.ra.hle

323

identity) and use the formula (2.2) which shows that (ala) are the eigenvalues of the tensor defined by (2.19) since, due to the definition (2.17), the coordinates (ul) are the eigenvalues of the tensor L. Formula (2.20) follows by iteration from (2.19). Finally, identity (2.4)4 implies

aa = - tr(Ka-i • L), a

and (2.21) follows from (2.19). •

REMARK 2 .1 . A first consequence of Proposition 2.1 is that the functions (ala) are the principal invari ants of the tensor L interpreted as a linear operator. So that by using one of the formulae (2.19)-(2.21) we can build the basis (Ka) of the Stàckel system by knowing the expression of L with respect to any coordinate system. Hence, the tensor L plays the role of generator of the Stackel system. Due to Lemma 2.1, we note that a tensor L is a generator of a Stackel system by means of formulae (2.19)-(2.21) if and only if: (i) L is a conformai Killing tensor; (ii) the eigenvalues (ul) of L are real independent functions (so that they can be interpreted as coordinates) with distinct values; (iii) ali the eigendirections of L are normal (the orthogonal distributions are completely integrable) and the corresponding eigenvalues (ul) are Constant on the integrai manifolds. The requirement "real" is automatically satished in a proper Riemannian manifold (that is for positive definite metrics). Tensors having properties (ii) and (iii) are characterized by the vanishing of the Nijenhuis tensor [NI].

REMARK 2.2. It can be seen that the assumption (2.16) for the metric tensor implies Rjj = '0 for i 7̂ j . This means that the coordinate curves are tangent to the Eicci principal directions. This is a necessary and sufficient condition for an additive separable orthogonal system (for the Hamilton-Jacobi equation) to be multiplicative separable for the Helmholtz and related equations. It is called the Robertson condition [RO] [KM1] [KA].

3. Inertia tensors and the orthogonal separation

Let M = {(-P4, mL) G ( E n , R); 1 — 1 , . . . , N} be a finite system of mass points in the Euclidean n-dimensional space. The masses mL can be either

324

positive or negative numbers. We denote by m the total mass, the sum of ali masses. If m ̂ 0 we can defìne the center of mass G through one of the equivalent equations

(3.1) Y,m>)2, m £ 0, LP(v) = LQ(V) + 2 PQ • v w • v, m = 0.

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PROPOSITION 3.1. The planar ìnertìa tensor L is a conformai Kìllìng tensor.

Proof. By defìnition, a symmetric 2-tensor fiele! L is conformai Killing tensor if along ali geodesics

(3-9) *EjL = Fci?,.

where C is a vector fìeld and

FL = \l>ij J *, Fc = Civi,

being (V) the components of the vector tangent to the geodesics. Along any curve P(t) in E n we have

i

If the curve is a geodesie curve (that is, a straight line), then. ^ = 0, so that

—- =.-C -vvz. dt

where C is the vector fìeld defìned by

V J r ^ \ Cp = - 2 w = const. (m = 0)

This proves the statement. •

The vector fìeld C is a gradient of a function / (we say that L is an exact conformai Killing tensor):

U{P)=\mr\ r = GP, \ f(P) = — 2 w • r, r = OP (0 = arbitrary point).

PROPOSITION 3.2. The ìnertìa, tensori is a Killing tensor.

326

Proof. We have two simple proofs. (i) A Killing tensor, on a Riemannian manifold, is a symmetric 2-tensor K such that the quantity

FK(P, «) = \KP{V)

is Constant along every geodesie curve P(t), where v is the unitary vector tangent to the geodesie. In the Euclidean space the gepdesics are the straight lines and moreover, due to the definition (3.7) of the moment of inertia, the quantity Fj(P) does not depend on the choice of the point P over a line, (ii) We have

di n \-^ ^ ^ 9 dL — = - 2 > mL PPL • v v

2 - —- = 0. • di ^ di

t

These propositions and the fact that equations (3.6) iìt with the step a = 1 in the iterative formula (2.18), with K\ = / , suggests that thè planar inertia tensor L is a good candidate for playing the role of generator of a Stàckel system of the kind considered in the preceding section.

4. Elliptic Stàckel systems

Let us consider the conformai Killing tensor L defined by (3.8) in the case m ^ 0 ,

(4.1) L = LG + mr®r (r = GP).

Let (xa) be Cartesian coordinates with origin O — G deiined by the eigendirections of LQ. Let (aa) be the eigenvalues of LQ and (u

l) the eigenvalues of L at a generi e point. Equation (4.1) is equivalent to

(4.2) La^ = aa6ap-\-mxaxp.

Let (Xa) be the unitary Constant vector fields corresponding to the Cartesian coordinates (xa).

PROPOSITION 4 .1 . If ai < a2 < . . . < an, then the eigenvedu.es (ul) of

L coincide with the elliptic coordinates a,nd the tensor L generates a, basis of the elliptic Stàckel system according to Proposition 2.1.

http://eigenvedu.es

327

Proof. Let us consider the vector fìelds

1 n

(4-3) Ei = ì

328

If we multiply by À — aa and evalùate the result for X = aaì then we get equations

1 0 a A'(aa)'

which relate the Cartesian coordinates (xa) with the elliptic coordinates (V) . From (4.7) it follows that

Oiiz 2 ul — aa^ and we iìnd the components of the vector fields (E{). Hence, the tensor fìeld L satisfi.es the requirements of Proposition 2.1. •

REMARK 4 .1 . The relevance of a tensor like (4.1), in connection with elliptic coordinates, is pointed out by Moser [MOl] [M02] within a different framework.

REMARK 4.2. According to Proposition 4.1, the roots of equation (4.4) are the eigenvalues of X, thus equation (4,4) is equivalent to the characteristic equation of X, so that the elementary symmetric functions (aa) of the roots (ul) are the coefficients of this equation and coincide with the principal in.varian.ts of L. The comparison between the characteristic equation of L and equation (4.4) shows that

n

(4.9) aa = qa + m ^ $2-1 xi (a = 0 , 1 , . . . ,n) ,

7=1

where (

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Proof. We can apply one of the formulae (2.19)-(2.21). Let us choose for instance the iterative formula (2.19). For a = 0, from (11) we get KQ01 = 1 and KQ = 0, that is KQ — #, in agreement with (2.19). Assume that the expressions (4.11) hold for the index a — 1 and let us check if they hold for the following index a. According to (2.19) we have,

7

• = Sa + m J2 $-ix* + m2 Y, ^O-VT X° 7 7^a

tf-l + mJ2 ^a-2xy 1 (a« + m xl)

= Sa ~ a a C l + m Y^ (

330

REMARK 4.4. Let us denote by (pa) the momenta correspondìng to the Cartesian coordinates (xa) and set

(4.14)

Then the integrala

correspondìng to the Killing tensors (Ka) are

la/3 = XaP(3 ~ *p pa .

1 T,af3

(4.15)

V*=l a ,13=1 )

R E M A R K 4 . 5 . If we defìne

_ n - i

(4.16) Ka=J^^Ka (a = l , . . . ,n) , a=0

where ( — •tr' a« — a7

= — a;2

-1» = — 77Ì , aa - a^

Kaì* — 777 xa$(3

i aa — G/3

K01 — 0 7

(a,/3,7 ^)

331

and the corresponding integrals

For m = — 1 they correspond to the involutive integrals found by Marshall and Wojciechowski [MW] (with Constant potentials).

R E MARK 4.6. If the coefficients (aa) are not ali distinct, then the Killing tensors (Ka) are well defìned but not ali independent, wliile some of the Killing tensors (Ka) are not defìned (however, those which can be defìned by (4.14) are independent). Under the assumption that ali (aa) are distinct, ali the Killing tensors (A r a), with the exception of KQ = g, have distinct eigenvalues

Qai — ^ai

which are functions of the coordinates (ul). The eigenvalues of the tensors (Ka) are the functions

n-l

(4.21) .5«=X«Sff«

of the 2n variables (ul,aa).

5. Parabplic Stàckel systems

Equation (3.8) for m — 0 shows that

(5.1) L = LQ + 7*® w+ w0r (r = QP),

where Q is any fixed point of the space. We assume tu . / 0.

PROPOSITION 5.1. There exists a unique point 0 sudi that LQ-W = 0. The vector w is an eigenvector of Lp for every point P of the straight line parallel to the vector w and containing O.

332

Proof. Due to (5.1), condition LQ • w = 0 is equivalent to

LQ • w + w2 r -f r • w w = 0, r—QO.

This is a linear equation in r which implies

i r

thus 1 /W-LQ-W

r = ~T — ^ 2 — W-LQW

This defìnes a unique point 0. Furthermore, let us consider the points P = O + kw where fcEl. According to (3.8),

Lp — LQ + 2fc w tu,

thus

since Z Q • w = 0. •

According to formula (5.1) and Proposition 5.1 we can write

(5.2) L = L0 + r®w + w®r (r = OP), Lo-w = 0.

Let (xa) be Cartesian coordinates with origin at the point 0 defined by the eigendirections of LQ. Let x\ = x be the coordinate corresponding to the eigenvector w. Let (aa) be the eigenvalues of LQ and (u

1) be the eigenvalues of L at a generic point. Equation (5.1) is equivalent to

(5.3) L01^ = aa 6ap + xa wp + xp wa,

where, according to Proposition 5.1,

(5.4) «i = 0, wa = Sia w.

Let (Xa) be the unitary Constant vector fìelds corresponding to the Cartesian coordinates (xQ), with X\ — X.

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PROPOSITION 5.2. If a2 < «3 < . . . < an, then the eigenvalues (V) of L coincide with the parabolic coordinates and the tensor L generates a basis of the parabolic Stackel system according to Proposition 2.1.

Proof. Let us consider the vector helds

(5-5) £- I(i*+£_4SL_JU, V i in £—4 il* — n _ 1 v a=2 "

n 2

w-here (w1) are the roots of equation

_xl 1 A

It follows that

334

where

n n (5.8) U(\) = fJ(A - !*•), A0(X) =l[(\- àa).

i= l a=2

Equations

(5.9) x = — [ V i i 1 : - V a a | , xl = A,\a[ (a = 2, . - : . ,n) ,

relate the Cartesian coordinates (a;0) with the parabolic coordinates (ul).

Equations (5.9)2 follow from (5.7) multiplied by A — aa and evaluated for A = aa. Equation (5.9)i is a particular case of formula (5.11) below (Remark 5.1), and it follows from the expression of the coefRcient of A n _ 1 in the algebraic equation equivalent to (5.6). From equations (5.7) it follows that

,- i n x OX 1 OXa 1 Xa (5.10) —r = — , _ = _ _ , ( a = 2 , . . . , ra)

dul 2w dul 2 ul - aa so that we get the naturai frame (5.5). Hence, the tensor fìeld L satisfìes the requirementsof Proposition 2.1. •

R E M A R K 5.1. Às a consequence of Proposition 5.2, since the roots of equation (5.6) are the eigenvàlues of L, equation (5.6) is equivalent to the characteristic equation of Z, so that the elementary symmetric functions (

335

REMARK 5.2. The defìnition of the parabolic coordinates (ul) depends on the differences aa — a a only, i.e. it is invariant under a translation over the real line of these constants (as for the elliptic coordinates). However, it is convenient to assume that aa ^ 0 for ali a > 1, so that ali the constants aa are different (silice a\ = 0).

PROPOSITION 5.3. The parabolic Stackel system corresponding to the distinti constants ( a 2 , . . . , a n )

i S generated by the Killing tensors (Ka) (a = 0 , 1 , . . . , n — 1) whose Cartesian components are

K i l

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REMARK 5.3. In the Cartesian frame (Xa) the Killing tensors (5.12) can be written as follows,

n n 1 n

(5.13) Ka = £

337

where A

fi = — . m

Keeping fi fixed, when m —> oo the ellipsoids in equation (6.2) become spheres (with r2 = fi).

A basis (Ha) of the spherical-elliptic Stàckel system can be directly derived from the elliptic one (Ka) defmed in (4.11) by considering only the terms proportional to the parameter m (this means that we consider the elliptic coordinates with large values of m). The parameter m does not appear in Ko = g, so that we take HQ = g. The other Killing tensors are

Ha = Ka - Ka{p) (a = l , . . . , n - l ) .

Thus,

PROPOSITION 6.1. The spherical-elliptic Stàckel system is generateci by the independent Killing tensors

1 n

(6.3) Ho = g, Ha = - ^ C-1 R

338

relate the Cartesian coordinates (xa) with the spherical-elliptic coordinates («•) [MOl]. They follow from the identity

xl _ r2 UoW n 2

(6'7) S^--^r Equations (6.5) imply

/ n 0 \ UXQI 1 # a . . . Oxa xa

ou% 2 ul - aa oun r

Thus the frame (E{) corresponding to the spherical-elliptic coordinates is

(6.9) Ei = \-p^-Xa (i

339

and a vector v = (va) is tangent to the sphere if and only if v • r = vaxa = 0. Thus

(H1.v)a = (l-x2a)v

a- JTxaxyvi

= (1 - x2a)va + x2av

a = va

that is, Hi • v = v. We note that the spectrum of the Killing tensor Hi is ( 1 , . . . , 1 , 0 ) .

R E M A R K 6.3. From Remarks 6.1 and 6.2 it follows that the n — 1 Killing tensors (Hi, Hi, ... , iTw_i) are reducible to every sphere centered at the origin and give rise to a Stàckel system which corresponds to the elliptic coordinates (us) = (u1,... , ^ n _ 1 ) originally considered by Neumann [NE] on the sphere.

R E M A R K 6.4. Within the framework present here the following further topics can be developed: (i) The construction of ali Stàckel systems in the Euclidean affine spaces. To this end it is necessary to discuss the case in which some of the constants (aa) coincide, (ii) The Cartesian characterization of ali the separable dynamical systems in the Euclidean affine spaces. To this end it is necessary to discuss the geometrical characterization of the orthogonal separation in a Riemannian manifold. This will be done in subsequent papers.

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Sergio BENENTI Istituto di Fisica Matematica "J.-L. Lagrange" Via Carlo Alberto 10, 10123 Torino, Italy.

Lavoro pervenuto in redazione il 2.11.1992.