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PUBLICATIONS

NAT'L INST. OF STAND & TECH

AlllDb 4D7721

NBSIR 82-2541

Normal Form and Representation

Theory

MathenatiSCh Instiluut, The Netherlands

Gemer tor Applied Mathematics

U.S. DEPARTMENT OP COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Applied Matnematics

Wasnington, DC 2.023.4

June ; 982

Final Report

U.S DEPARTMENT OF COMMERCE

National Bureau of Standards

^QC'""" nal Engineering Laboratory

2QQ ir for Applied Mathematics

.1156

82-2541

1932

02

inqton. DC 20234

51 ,'norJAIi BTJRttAU O/ BTANT)AJU33

LtWR.' ’ V

NBSIR 82-2541

AUG 1 9 1992

noV<XC£L " ^

Q.C l°°

NORMAL FORM AND REPRESENTATION

THEORY 5>

2.

Richard Cushman*, Andre Deprit**

and Richard Mosak**

*Mathematisch Instituut, The Netherlands

**Center for Applied Mathematics

*Mathematics Instituut

Rijksuniversiteit Utrecht

Budapestlaan 6

Utrecht, The Netherlands

**U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Applied Mathematics

Washington, DC 20234

June 1 982

Final Report

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Applied Mathematics

Washington, DC 20234

z

Q o

Q

*

-r

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

1

Normal form and representation theory

Richard Cushman

Mathematisch Instituut, Rijksuniversiteit Utrecht,

3508 TA Utrecht, The Netherlands,

Andre Deprit and Richard Mosak a )

National Bureau of Standards, Washington, D. C. 20234, U.S.A.

( Received
2

Representation theory of Lie algebras is called upon to

develop a procedure for normalizing a dynamical system with two

degrees of freedom in the neighbourhood of an equilibrium when

the Hamiltonian H(x, y, X, Y) in the coordinates (x, y) and

their conjugate momenta (X, Y) is of the type

H = (X

2

+ Y

2

)/2 + V(x, y, X, Y) , the potential energy V being a

sum of homogeneous polynomials in the phase variables of degree

strictly greater than two. The fact that the resulting

potential V' is a polynomial in the new coordinates (x f , y’)

and the angular momentum G’ = x' Y' - y

f X

T implies that the

normalization is a rotation in the configuration space from a

fixed frame to an ideal frame. The technique is intended for

normalizing an Hamiltonian in equilibrium at the origin when

the Lie derivative associated with the quadratic part is not

semi-simple, e.g. the planar Restricted Problem of Three Bodies

at the equilateral equilibrium L4 when the basic frequencies

are equal (Routh’s singular case).

PACS numbers: 03.2CH-i, 46.10+z, 95.10Ce
3

1. INTRODUCTION

The literature about normalization deals mainly with semi-simple systems

in equilibrium at the origin. The Hamiltonian being a formal series

H i H(x, y, X, Y) I “ H„

n>0 n!

( 1 )

whose terms Hn are homogeneous polynomials of degree n + 2 in the coordinates

(x, y) and their conjugate momenta (X, Y) , it is generally assumed that the

dominant term Hq is a quadratic form reducible to the type

J = - (X2 + ei uq 2 x

2

) + - e (Y2 + £2 ^ 2‘

Y

2

) » ( 2 )

in which the frequencies ojj and 002 are real and > 0, the factors e, and £2

being either +1 or -1. Such systems are called semi-simple because their

dominant term leads to a linear Hamiltonian vector field that is semi-simple.

Let

Lj : F ^ Lj(F) = (F; J)

be the Lie derivative associated with the Hamiltonian linear vector field

derived from J. For any n ^ 2, the restriction of the differential operator

3 3 3 3 Lj = (X — + £i co i 2 x —) + £ (Y — + £2 <*>2 2 y —

)

8x 3 X 3 y 3Y
4

to the vector space Pn of homogeneous polynomials of degree n in (x, y, X, Y)

is an endomorphism of Pn that is semi-simple; hence, with respect to Lj, Pn may be decomposed into the direct sum

Pn

= Im Lj + Ker Lj . (3)

The concept of normalization for semi-simple systems in equilibrium at

the origin must be credited to Whittaker^- who created it first by adapting a

method proposed by Delaunay^ for eliminating periodic terms from the main

problem of Lunar Theory. Later, Whittaker^ carried out the normalization as a

canonical transformation (x, y, X, Y) - (x' , y’ , X', Y

f ) defined through the

implicit equations

3 S 3 S 3 S . 3 S X = — , Y = — > x

f = » y' = 3x 3y 3 X

f ' 3Y T

derived from a generating function

S = S(x , y , X* , Y

f ) = l

S n n^O

where Sq = x X' + y Y' and, for any n > 1, the term S n

is a homogeneous

} ) f polynomial of degree n + 2 in (x, y, X’, Y’). Although devised by Poincare 4

as one of his " methodes nouveiles "

, the procedure is referred to in some

quarters of celestial mechanics as von Zeipel's method. Nowadays^ the

normalization is executed as a Lie transformation^ in the sense of to convert

the formal power series (1) into a formal power series
5

(4)

such that (i) H’q = J(x r , y' , X’, Y* ) , and (ii) for each n > 0, H’ n belongs to

the kernel of Lj. In action- and angle- variables ($ , ip

, $ , f ) , a

polynomial in (x, y, X, Y) becomes a trigonometric sum in (<j> , ip), its

component in Im Lj consists of short-period terms whereas its component in Ker

Lj groups the terms which are either secular or of long period. The

normalization is justified in this framework as a technique for removing

short-period effects from the perturbation. The requirement that

(H f ; J) = (H'o; H

f ) =0 implies at once that the dominant term H'q as a

function of the normalizing variables is a (formal) integral of the system,

hence calls for a reduction of the Hamiltonian system from two to one degree

of freedom.

Whittaker ? in his theory of Integration by Series, and most textbooks

following him, considers exclusively semi-simple Hamiltonians in equilibrium.

It must be realized® though that a non-degenerate quadratic form in

(u, v, U, V) is reducible by a real symplectic linear transformation

(u, v, U, V) + (x, y, X, Y) either to type (2) or to the type

1 2 (x2 + y

2 J ) = id (xY - yX)

2

(5)

In the second case, the Lie derivative

Lj = u) (x — - y —) + to [ ( Y - eajx) — - (X + ewy) —

1

3 X 3 Y 3 X 3 Y

( 6 )
6

will be decomposed into the sum

Lj = co Lq + s m

2 Lq

where the differential operators

3 3 3 3 L_ = x — - y — + X — - Y — » 3y 3x 3 Y 3X

tD = x

3 3 — + y —

3X 3 Y

( 8 )

(9)

are the Lie derivatives corresponding to the Hamiltonian linear vector fields

derived from the functions

G = xY - yX and

1 9 D (x2 + y

2

)

.

2

( 10 )

The sum (7) realizes a Jordan decomposition of the endomorphism Lj : Pn * ?n>

that is, is semi-simple, Lp is nilpotent, and these operators commute since

3 3 k> tG - tG ld - * ~ - y - ( 11 )

For any n > 0, the vector space Pn turns out to be the direct sum

?n = Im tG + Ker y 900
7

and, on account of the commutativity relation (11), the restriction of LD to

the kernel of LG

is an endomorphism of Ker LG. Hence the normalization of a

non semi-simple system in equilibrium at the origin could proceed in two

steps. First a Lie transformation <p : (x, y, X, Y) (x'

,

y* , X', Y')

normalizes the system with respect to the semi-simple component L^, thereby

changing (1) into a power series (4) in the kernel of Lq. In the new

phase variables, the angular momentum G ' = x’ Y

f - y' X’ is a (formal)

integral; hence the term aj G' may be omitted from the dominant part in the

transformed Hamiltonian, then reduced to J' = e w

2 D’ . Rather than analyzing

the partially normalized problem as a system reduced to one degree of freedom

by means of the integral of angular momentum, van der Meer^ proposes that the

normalization be continued with a Lie transformation \J; : (x* , y* , X’, Y’) +

(x", y"

,

X"

,

Y”) to convert (4) into a formal power series confined to a

remarkable vector subspace of Ker Lq . Indeed, relative to the Lie derivative

( 12 )

associated with the Hamiltonian vector field derived from

( 13 )

the kernel of Lq may be decomposed into the direct sum

Ker Lq = ( Im Lq Ker Lq ) + ( Ker Lr Ker Lg )

.
8

This makes it possible for the transformation \J; to convert ( 4 ) into a (formal)

power series

H" = H" (x" , y", X", Y”) = l — H" n n>0 n!

that belongs to Ker Lq Ker Lg* As a combined effect of the transformations

and 41, the kinetic energy H"q = e co 2 D” and the angular momentum G" in the

third set of variables come out formally as integrals. Because they admit two

independent integrals in involution, non semi-simple systems with two degrees

of freedom in equilibrium at the origin are ( formally ) integrable .

van der Meer did not concern himself with developing an algorithm for

generating the second normalization. As for ourselves, while engaged in

designing such a procedure, we noticed that our techniques apply to a class of

«

* systems wider than the nilpotent part of a non semi-simple system in

equilibrium at the origin. In fact exchanging the coordinates and the momenta

transposes the Hamiltonian of the latter system into one whose dominant term

H’q is K' instead of D' . Such Hamiltonians represent motions of a particle

subject to weak perturbations in the neighbourhood of the origin. In that

general context, the main contribution of this article is summed up in the

following
9

Theorem Given a Hamiltonian system with two degrees of freedom

S \ VV N \ S ' - — ' — — — — — " - - — —. —

1 represented by the formal series

1 „ „ r* 1 H = H(x, y, X, Y) = - (X2 + Y

2

) + V — Hn

(x, y, X, Y) (14)

2 n5l n!

where , for n > 1 , the perturbation ^ is_ a_ homogeneous polynomial of degree

n + 2, one may build formally a Lie transformation (x, y, X, Y) + (x f , y\

X*

,

Y

f ) to convert H into a formal power series

H' = H'(x’, y' , X’, Y’) = - (X’ 2 + Y’ z

) +

•» 2

' 1 n5 1 n!“ H' n (x', y', X’ , Y’)

where , for each n > 1 , H' n^Ln the kernel of Lp. More precisely ,

H'n = a+6+2y=n+2 Ve,Y x'<VBg'Y .

The physical meaning of the normalization is exposed below in the

corollary to the theorem. For the resulting potential

n 1 V' i V'(x', y', G* ) = I — H' n

(x'

,

y\ G’)

n5l n!

of the forces acting on the particle in the neighbourhood of the origin, let

the differential be written as the 1-form
10

dV* = 3^’ dx’ + 3 2 V

f dy' + 83V’ dG' .

Also, let (C) be the original Cartesian frame of reference in the

configuration plane (x, y) with i and | standing for the orthonormal unit

vectors in the directions of the reference axes. Finally let (I) designate the

moving frame obtained by rotating (C) about the origin at the adjusted angular

velocity -83V’, i* and j’ denoting the orthonormal unit vectors in the

directions of the coordinate axes. In these notations

Corollary .- The particle * s position x, its velocity v and its

acceleration a relative to the frame (C) are such that

Thanks to the normalization, the motion in the frame (C) is decomposed

into a rotation about the normal to the plane at the variable rate

-

33V' (x' , y' , G'), and a motion with respect to the moving frame (I). In the

latter frame, whereas one would have expected Coriolis forces and centrifugal

repulsion to compensate for the slow rotation, one finds that the forces are

reduced to the gradient of the force function U* = -V. Thus the moving frame

(I) constitutes what Hansen calls an ideal frame, and the normalization may be

regarded as a procedure to extract from Hamiltonian (14) the instantaneous

rate at which, along each particular orbit, the frame (C) should be set in

x = x i + y j = x ' i ' + y ’ j

' ,

v = X i + Y j = X' i'+Y' y ,

(15)

(16)

a = - 3 !

V

f i’ - 3 2 V' . (17)

rotation so that it becomes the ideal frame proper to that particular orbit.
11

Symmetry Lie algebras provide a natural framework in which to consider

the normalization of Hamiltonian systems. A case in point is the class of

semi-simple systems in equilibrium which admit a 1:1 resonance. Credit for

having discovered there the relationship between symmetry Lie algebras and

normalization goes to Kummer^. As matter of fact, an algorithm can be

set up to produce immediately the reduced Hamiltonian as a function over the

Lie algebra su(2) spanned by the symmetry generators H . In that light,

normalization of semi-simple systems in 1:1 resonance is but an application of

the Reduction Theorem. 12 a similar situation occurs for perturbed

two-dimensional Keplerian problems where the Delaunay normalization^ builds

the reduced Hamiltonian as a function over the Lie algebra so(3). The present

article shows that, for systems of type (14), the connection between symmetry

Lie algebras and normalization is equally decisive, but of a different nature.

For the symmetry Lie algebra is here solvable — and not semi-simple. However

the Lie derivative L{^ may be embedded in a simple Lie algebra, namely

sl(2, R)

,

whose representation specifies the requirements imposed by the

normalization.
12

2. SYMPLECTIC SYMMETRIES

The task of developing a normalization algorithm for nilpotent systems of

type (14) when K is given by (13) begins with studying the linear

infinitesimally symplectic symmetries for the Hamiltonian K. The Lie algebra

sp(4, R) of all infinitesimally symplectic linear maps (x, y, X, Y) +

(x'

,

y*

,

X’, Y’) is isomorphic to the Lie algebra of quadratic Hamiltonians

under the Poisson bracket^: to the infinitesimally symplectic matrix w in

sp(4, R) corresponds the quadratic form

where

W (x, y, X, Y) = - ( Jw v, v )

0 I J = and v = (x, y, X, Y),

-I 0

in which case W is said to generate the infinitesimally symplectic linear map

w. The latter is called a (linear infinitesimally symplectic) symmetry of K if

there is a scalar v such that [w, k] = v k if k is the infinitesimally

symplectic linear matrix generated by K, or equivalently

(W; K) = v K .

The definition adopted here for a symmetry is borrowed from Cartan^; it

allows for a reparametrization of the integral curves of the vector field

O0O

derived from K by the symmetries generated from W.
13

A quick evaluation shows that the infinitesimally linear symplectic

symmetries form a five-dimensional solvable Lie algebra, a basis of which is

generated by the sum

W = ej + £2 ^2 + e.3 ^3 + e 4 ^4 + e 5 ^5

whose terms are Wj = K, W3 = G,

W2 = S = xX + yY, W4 =

l-

( X

2 - Y2

) , W5 = XY .

In G, one recognizes the generator of the rotations about the origin. From the

infinitesimal symmetry s defined by the equations

x' = x , y’ = y , V = - X , Y' = - Y

derived from S, the exponential mapping e es produces the symplectic symmetry

x’ = e

e X , y* = e

e Y , X

f = e"e X , Y' = e“e Y

in finite form. It corresponds to the similarity due to the homogeneity in

dimensions : multiplying the coordinates by a constant X (= e

e

) and dividing

the velocities by the same constant requires, for preserving the dimensional

2 homogeneity, that the time itself be multiplied by X . This rule of similarity

can also be verified by checking that the symplectic symmetry changes Cartan's

form
14

a) = X dx + Y dy - K dt

into the 1 -form

a) = X’ ax’ + Y' dy' (X * 2 + Y

f

2

) dt ,

thus suggesting that the time t be replaced by the independent variable t' =

X

2 t.

The symmetry generators K, G, W4 and W5 are evidently integrals for the

free particle. The function K and G are respectively the particle's energy and

its angular momentum; the integrals W4 and W5 restate the principle of

conservation of linear momentum since they combine to yield that X + Y and

X - Y, hence the components X and Y of the velocity, are integrals. The

generator S itself is not an integral of the free particle. However the

symmetry condition (S; X) = 2K gives rise to the differential relation S =

(S; K) = 2 K, and thus to the integral S - 2 K t = cj_ from which, by yet

another quadrature, is derived the so-called Jacobi integral

D + S t - K t

2 = cq .

The infinitesimal symmetries w-j_ derived from the generators Wj_

(1 < i < 5) span a subalgebra in the linear Lie algebra sp(4, R) of 4 x 4

symplectic matrices. The family is completed into a basis w^ (1 < i < 10)

TABLE

I of sp(4, R) . Table I lists the generator and the non zero elements w-^j^ (in

row j and column k) for each symplectic matrix w^ in the basis.
15

For the basis of sp(4, R) established in Table I, one computes easily (at

least on a home computer) the commutators [w-^, wj ] = w-£ wj - wj ( 1 < i,

j < 10) , the results of which have been entered in the respective i-th row

TABLE

II and j-th column of Table II.

Basic to the normalization are the following facts which can be read

immediately from Table II of the commutators.

a) Each element a of a Lie algebra A over a field F determines an

endomorphism ad a : b + [a, b] : A -*• A of the F-vector space. The element a is

said to be ad-nilpotent if (ad a) m = 0 for some integer m > 0. A quick

3 calculation using Table II shows that (ad wj) =0, or that the symmetry w^ is

ad-nilpotent in the linear algebra sp(4, R)

.

b) In a finite-dimensional semi-simple Lie algebra A over the real or the

complex field, every ad-nilpotent element may be embedded in a subalgebra B of

A isomorphic to the special linear Lie algebra sl(2, R) of 2 x 2 matrices with

zero trace^. More precisely, there exists a pair (b, c) of elements in A such

that

[a, b] = 2b , [a, c] = -2c , [b, c] = a .

In application of the embedding theorem to the ad-nilpotent matrix w^ in

sp(4, R) , one reads from Table II that

[w2 , wj] = 2w i , [w£, wg] = -2wg , [wj, wg] - W 2 . (16)
16

c) Each matrix w-^ in Table I is, by construction, the Hamiltonian linear

vector field derived from the corresponding generator But, given two

vector fields a and g, the Lie derivative of their commutator [a, g] is

Hence the embedding relations (16) transposed in terms of Lie derivatives

yield

The lemma could have been obtained by evaluating long-hand the

commutators of these Lie derivatives from their expressions as differential

operators, or by evaluating the Poisson brackets (W-j_; Wj ) since

While an algebraist interprets the results in Lemma 1 as saying that L^

and Lq are eigenvectors of ad Lg with eigenvalues respectively equal to 2 and

-2, a physicist reads them as meaning that Lg is a symmetry of the vector

field derived from K that shortens the time along the integral curves of K

while, as a symmetry of the vector field corresponding to D, it slows down the

usually^ defined as the differential operator

t[ag] tgl “ tg I?a “ ta ^g *

[L , L ] = L - Wi -Wj " = L

(Wi; Wj) ~[wj

} w± ]

time along the integral curves. This interpretation brings forth a close

analogy between the normalization proposed in this article for perturbed free

particles and the conventional averaging procedures applied to conditionally
17

periodic systems: in both cases, the algorithm removes the short term effects

caused by the perturbations.

From Tables I and II, the reader may collect the following triples:

(xX; - X

2

) = X

2

, 2

1 9 9 (yY; - Y

2

) = Y

2

, 2

(xX+yY; XY) = 2XY ,

(xX-yY; Xy) = 2Xy ,

1 9 9 (xX; x

2

) = x

2

, 2

(yY; - ; Y

2

) = y

2

, 2

(xX+yY; -xy) = 2xy ,

(xX-yY; xY) = -2xY ,

1 9

1 9 (- X

2

; - - x

2

) = xX ; 2 2

(“ Y

2

; - ;

y

2

) = yY ; 2 2

(XY; - xy) = xX+yY ;

(Xy; xY) = xX-yY .

As will be seen in the next sections, to each of them corresponds a

representation of the simple algebra sl(2, R) in the general linear algebra

gl(Pn ) relative to the vector space P n , hence a normalization scheme for a

certain class of Hamiltonians. For example, the first triple concerns

Hamiltonians of the type

1 „ „

1 H = H(x, y, X, Y) = - X

2 + 7 — Hn (x, y, X, Y)

2 nSl n!

which may be normalized into Hamiltonians

1 o r1 H' = H'(x', y', X’, Y’) = - X' 2 + 7 -- H

, n

(x’ , y' , Y')

2 n£ 1 n

!

with the momentum X' eliminated from the perturbations.
18

3. DECOMPOSITION OF THE PERTURBATIONS

The objective of this section is to prove that any homogeneous polynomial

p may be written in a unique way as the sum P = Pk + PD °f two homogeneous

polynomials of the same degree such that Lp pp = 0 and p^ = q for some

homogeneous polynomial q. It is to be noted that, in this section, and Lp

stand respectively for the restrictions of Ljr and Lp to the vector space ?nThe decomposition leads to a procedure for constructing a Lie transformation

which will strip any perturbation term Hn in (14) of its component in Im L^.

Actually the decomposition of Pnas the direct sum of Ker Lp and Im

is but a particular case of a more general result about the representations of

the special linear Lie algebra sl(2, R) . To get at the essence of the problem,

we need however to fix notations and terminology, and to recall a few

basics from Representation Theory.

Let V be a real vector space of finite dimension; in the algebra of

endcmorphisms of V, [a, b] denotes the commutator ab - ba. For readability,

the image b(x) of a vector x in V by an endomorphism <j> will be written simply

as <jjx. One says that V is an sl(2, R) - module or, equivalently that sl(2, R)

is represented on V, if there exist three non-zero endomorphisms x, y, h of V

satisfying the relations

[h, x] = 2x, [h, y] = -2y, [x, y] = h. (17)

By virtue of Weyl ' s theorem, an sl(2, R) - module V may be decomposed into a

direct sum of real vector subspaces
19

V = Vj + V2 + ... + Vs (18) •

whose summands are invariant and irreducible under the set of endomorphisms

(x, y, h) . There may be more than one way of accomplishing the decomposition,

but the number s of summands and the equivalence classes of the irreducible

representations are uniquely determined.

Proposition 1 .- Relative to a real vector space V of finite dimension SVSVSSVSVNS V I ~ -i —— - — — - — - - I

that is an sl(2, R) - module for the three endomorphisms x, y, h satisfying

the commutator relations ( 17)

,

(i) the endomorphisms x and y are nilpotent ;

(ii) the endomorphisms xy and yx are semi-simple ;

(iii) Ker xy = Ker y and Im xy = Im x; likewise , Ker yx = Ker x and

Im yx = Im y;

(iv) V = Ker x + Im y; likewise , V = Ker y + Im x; o

(v) dim Ker x = dim Ker y

= s where s is_ the number of summands in any

decomposition of V into a_ direct sum of irreducible vector subspaces .

The proposition is proved in two stages. First is considered the

particular case when s = 1, that is, V itself is irreducible; then the results

are extended to the general case where s > 1.

When it is irreducible, V admits,

(v0 , VJ, ..., vm) such that, for 0 < i < m

according to Humphreys^,

y

a basis

hv-L = (m-2i) vi. > (19)

yvi = (i+1) vi+l > (20)

xv-£ = (m-i+1) vi-l > (21)
20

with the convention that v_^ = v^^ = 0. As a consequence, xm“^Vi = ym~l

= 0, which proves that the endomorphisms x and y are nilpotent. There follows

also that (xy)v-^ = (i+1) (m-i)vi; hence the vectors (v^) (CKi<m) are a basis of

eigenvectors for the endomorphism xy (and likewise for the endomorphism yx)

,

which means that xy and yx are semi-simple. For that reason,

V = Ker xy + Im xy, and also V = Ker yx -r Im yx . More precisely, since,

on the one hand, (xy)v^ = 0 if and only i = m, and, on the other hand,

Vj_ = (xy) [ (i+l)“^(m-i)“^Vi] for CK i < m-1, it turns out that Ker xy is the

one-dimensional vector subspace generated by vm , while Im xy is the

m-dimensional vector subspace generated by the vectors v^ (0 < i < m-1). In

view of this decomposition of V, statements (iv) and (v) in the proposition

are immediate corollaries of statement (iii). There remains thus to prove

point (iii). To this end, observe that, on account of relation (21), Vj[ is in

the image of x if and only if 0 < i < m, while xvm = 0; therefore, being the

vector subspace generated by the vectors v-^ (0 S i < m-1), Im x is identical

to Im xy. Similarly, by virtue of (20), a linear combination

ag vg + aj V 1 + • • • + am vm kernel of y if and only if a^ = 0 for

0 < i < m-1; hence, being the one-dimensional subspace generated by vm , Ker y

is identical to Ker xy.

The proof of Proposition 1 in the general case where V is completely

reducible although not irreducible rests on decomposition (18) of V into a

direct sum of irreducible submodules. It has just been proved that the

restrictions of x and y to each of the summands are nilpotent, and that the

restrictions of xy and yx are semi-simple; hence the endomorphisms x and y

themselves are nilpotent in V, whereas the endomorphisms xy and yx are

semi-simple in V. Demonstration of assertions (iii) - (v) involves the

following
21

Lemma 2.- Let V be a vector space that is a direct sum of the

S \ S V S V — — — —— - — ' - —— — .... - - - — - — "

subspaces (V^) (1 < i < n). Then , for any vector subspace W of V, the

following statements are equivalent ;

(i) W is the direct sum of the vector subspaces (W V-^) (1 < i < n)

;

(ii) For any element w in W, the decomposition w = wj + W2 + . . . + wn

such that w-l belongs to V-£ for 1 < i ^ n implies that each component w-j_

belongs to W.

Intuitively speaking, the lemma says that W is the direct sum of its

intersections with the V^'s if and only if the components of any vector in W

along every "direction" lie in W. Elementary as it may be, this lemma is

not mentioned in the major textbooks in Linear Algebra; it is therefore in

order to skecth its proof. Considering that V is the direct sum of the

subspaces , any element w in W may be decomposed into the sum

w = wj 4- W2 + . . . + wn with wj_ in *V^ for each i. But, assuming that (i) holds,

the same element w may be decomposed into the sum w = u^ + U 2 + . . . + un with

u-?_ in W V-£ for each i. The decomposition of w in V being unique, there

follows that w-£ = u^, hence that w^ belongs to W for each i, which shows that

(i) implies (ii). Conversely, if (ii) holds, then w^ belongs to W for

each i. Such a decomposition being unique, there results that W is the direct

sura of the subspaces W V-^, hence that (ii) implies (i).

Lemma 2 is used to prove that

Im z = (Vj Im z) + (v 2 Im z) + ... + (v s Im z) , (22)

Ker z = (V : Ker z) + >(NJ Ker z) + ... + (vs Ker z) (23)

•oo

990

99«

00©

900

909
22

when the endomorphism z is either x, y, xy or yx. Indeed, take w in Im z; in

view of Weyl's theorem, it may be decomposed into a sum w = w^ + W£ + . . . + ws where w-^ belongs to V-j_ for 1< i < s. But w = zv for some v in V which in turn

may be decomposed into a Siam v = + V'l + . . . + v s with v± in V-^ for

1 < i < s; therefore zv = zv^ + zv£ + ... + zv s

. Now, since z leaves each

vector subspace Vj_ invariant and since the decomposition of w into its

components in the V^'s is unique, there follows that w^ = zvj_ for 1 < i < s,

or that each component lies in Im z, hence formula (22) on account of Lemma 2.

Similarly, for w in Ker z, there results that 0 = zw = zw^ + zw2 + ... + zws ,

hence that zwj_ = 0 for 1 < i < n, since 0 = 0 + ... + 0 is the unique way of

decomposing the null vector in the direct sum (18). Because each component w^_

belongs to Ker z,. formula (23) results also from Lemma 2.

Proposition 1 having been proved when V is irreducible, there follows

that Im xy|V-j_ = Im x|v^ and Ker xy|Vj_ = Ker y | for 1 < i < s. Then, by

reason of the relations (22)— (23) , one concludes that Im xy = Im x and

Ker xy = Ker y, which thus proves statement (iii) in Proposition 1. The next

statement Is then a consequence of the fact that the endomorphism xy is

semi-simple. Finally, since Ker x|v^ and Ker y|Vj_ are 1-dimensional for

1 < i < s, one concludes from formula (23) that Ker x and Ker y are of

dimension s. This completes the demonstration of Proposition 1.

Corollary - The vector space ?n is the direct sum In L[( + Ker Lp.

If V is an sl(2, R) - module, the real number X is called a weight of the

representation when the vector subspace of elements x such that hx = Ax is

not the null space. In Representation Theory, it is proved that the number s

of summands in the direct decomposition (18) is equal to dim Vq + dim V]^. In

•0O
23

the present case, the semi-simple endomorphism involved in the representation

of sl(2, R) is the differential operator

3 3 3 3 Lc = x — + y X Y — * 3x 3y 3X 3 Y

From now on, in order to disencumber the notations, monomials like xay^X^Y^

will be denoted e(a, 3, y, 5). In those terms,

Ls e(a , 3, Y ,<$ ) = (a + 3 - Y

- 3) e(a , 3, Y > 5 ) J

hence the monomials e(a, 3, Y> 3) constitute a basis of the vector subspace V^.

of Pnif and only if

a+3+Y+<5 = n and a + 3 - Y = k.

• 2 Therefore, when n = 2m, Vj_ reduces to the null space whereas dim Vo = (m+1) ;

otherwise, when n = 2m+l, it is Vq that is reduced to the null space while

dim = (m+-l ) (nH-2) . Hence

2 Proposition 2.- The dimension of Ker Lq in Pn

_is_ equal to (nri-1) when

n = 2m, and to (m+l)(m+2) when n = 2m + 1

.

I By consulting Table III, the reader will gain a measure of appreciation

for the extent to which the normalization simplifies the perturbed system

(14). An arbitrary polynomial that is homogeneous of degree n in (x, y, X, Y)

n+3

is the sum of ( ) monomials. Thus, from degree n to degree (n+L), the
perturbation tern Hn grows in complexity by (n+2)(n+3)/2 terms;

Ker Lq increases only by 1 + (n+l)/2 terms.

As the next section will indicate, there is however

normalization than a drastic reduction in algebraic complexity.

24

by contrast,

more to the
25

4. THE ANGULAR MOMENTUM INSIDE THE PERTURBATION

The decomposition vouchsafed by the corollary to Proposition 1 owes its

physical interest to the algebraic nature of the kernel of Lq as one can see

from

Proposition 3.- With a+B+2y = n, the polynomials g(a , 8, y) = x

ay^G^

form a. basis of Ker Lq in Pn , and the polynomials G(a , 8 , y ) = , a_ basis

of Ker L& i_n Pn.

The proof of Proposition 3 rests on yet another decomposition of P n into

a direct sum of vector subspaces which will be detailed first. In what

follows, most of the time, a monomial e(a, 8, Y, 5) will be identified with

the quadruple (a, 8, y, 5) of its exponents. Clearly, an arbitrary quadruple

(a, 8, y, 6 ) of integers represents a monomial if and only if its elements are

non-negative, in which case the monomial belongs to Pn if and only if

a + 8 + Y + <5 =n; let En be the set of all quadruples satisfying these two

conditions. The relation R defined by

(a', 8', y '

,

5’) R (a, 8, y

,

5

)

if there exists an integer k such that

(a’, 8', y’, <$') = (a, 8, y, 6) + (-k, k, k, -k)

is an equivalence among quadruples. For any quadruple (a, 8, Y, 5), let

E(a, 8, Y, 6) designate the intersection with En

of the class of quadruples

equivalent to (a, 8, y, 5) modulo R; given the integer k, the quadruple

(a-k, 8+k, y+k, 5-k; is an element of E(a, 8, Y> 5) if and only if

a-k > 0, 8+k > 0, y+k > 0, 5-k > 0
26

or, equivalently, if and only if

-min(B> y) < k < min(a, 5);

this shows in particular that the number of quadruples in the equivalence

class E(a, 8, y, 5) is equal to 1 + min(a, 6) + min(8 , y). The natural

ordering on the integers k provides an ordering on E(a, 8 » y ,

<5 ) . In that

order, the lowest element in E(a, 8, y, 5) is of the form (ag, 8g, T0> <$g)

with 8oyo = 0. Repeated addition of (-1, 1, 1, -1) to the lowest element

produces all the quadruples in the equivalence class E(ag, 8g> Y0> <5 o ) UP t0

the highest element which is of the form (cq, 8

1

> yj, 6]_) with a]_6i = 0.

Now let P(a, 8, y, 5) denote the vector space generated by those

monomials whose exponents belong to the class E(a, 8, y, 5). The monomials in

E n being a basis of P n , the preceding discussion establishes that Pn

is the

direct sum of the vector subspaces P(a, 8, y, 6), and that dim P(a, 8, y, <5 )

= min(a, 5) + min(8, y) + 1. However the vector subspaces P(a, 8, y, 5) are

not in general invariant under the operators and Lg; as a matter of fact,

the way in which these operators act on P(a, 8, y, 5) is given in the next

s tatement

.

Lemma 3 - For any (a, 8, y, <5 ) in. En and with the convention that

P( a, 8 , y , 5 ) designates the null vector space when E(a

,

8 , y ,

<5 ) is. empty ,

(i) Lk maps P(a

,

8

,

y

,

5

)

into P(a-1 , 8 , y+1 , 5 )

;

(ii) Lg maps P(a, 8, y, <5 ) into P(a+1, 8, y-1, 5);

(iii) LgL^ maps P(a

,

8

,

y

,

5

)

into itself .

Indeed, for any monomial e(a, 8, y, 5 )

,
27

Lr e(a, 3, y, 6) = a e(a-l, 3, Y+l , 5) + 3 e(a, 3~1> Y» *5+1) (24)

by virtue of definition (12). But (a-1, 3, Y+l, 5) = (a, 3~1, Y, 6+1)

+ (-1, 1, 1, ”1), hence both monomials in the right-hand member of (24) are

equivalent modulo R, and (i) is thus proved. By an analogous argument, (ii)

is a consequence of the identity

resulting from definition (9). Then (iii) follows by composing (i) and (ii).

Against this background information, Propsition 3 will now be proved, but

for only, since the result for Lp follows from swapping the coordinates

(x, y) and the momenta (X, Y) . Because is a derivation, the relations

Lj£ X = Y = (xY-yX) = 0 imply that G(a, 3, y) = 0; hence there remains

to show that the polynomials G(a, 3, y) are linearly independent and that they

span Ker L^. On the one hand, the binomial expansion

shows that G(a, 3, y) belongs to the vector subspace P(y , 0, a, 3 +y). The

equivalence classes E(y

,

0, a, 3 +y) and E(y '

,

0, a’, 3 '+y’) being disjoint
28

TABLE

IV

when (a, 3, y, 5) * (a ? , 3', Y

f , 5’), distinct polynomials G( a, 3, y) belong

to distinct vector subspaces P(y, 0, a, B+y), which means that the

polynomials G(a, 3, y) are linearly independent.

On the other hand, take a polynomial p in Pn , and let p = p x be its

decomposition relative to the subspaces P(a, 3, Y> <5 ) . Then p = p x ;

Since, according to Lemma 3, for i t k , the images L^ p x and L^ P< belong to

distinct subspaces P(a, 3, Y, 5 ) , the relation L^ p

= 0 implies that Lj( p x =0

for each index i . There results by reason of Lemma 2 that Ker L^ is the direct

sum of the vector subspaces Ker L^ P(a, 3, y, 5). As will be seen, all of «oo

these summands are identical to the null subspace save the intersections

Ker L£ P(y , 0, a, B'+Y ) which are of dimension one and are in fact generated

by a polynomial G(a, 3, y) (see Lemma 4 below). This will prove that such

special polynomials span the kernel of Lg;, and therefore constitute a basis of

# Ker L^ as is announced in Proposition 3. There remains thus to examine the

trace of Ker L^ on each vector subspace P(a, 3, Y, 6 ) , which will be done by

studying in detail the action of L^ on each of them. Such an ‘analysis will

prove useful also in Section 5 where the normalization algorithm will be

developed; it will show in particular that the decomposition of a polynomial

into its components in Im L^ and Ker Lg reduces to inverting a few matrices of

very low dimension.

Depending on the type presented by the lowest element (ag, 3g> Y0> 5g) in

P(a, 3, Y, 6

)

, the five cases mentioned in Table IV have to be considered. It

is a question of setting proper bases for representing the linear map L^ ’

P(a, 3, Y, 6) - P(a-1, 3, y+1, 5 ) as a matrix. To this end, in each case, the

monomials ordered from lowest to highest are adopted as a basis, the

coefficients of a polynomial are regarded as a column vector, and the operator
29

is represented as a matrix acting by multiplication to the left.

Class I can be disposed of at once. For it is clear that P(0, 0, yq> 5g)

Y 5 is of dimension 1, being generated as it is by the monomial X and that

it is mapped by onto the null space.

In class II, the monomials

e(aQ, 0, yq> $()), ***» e ^° 5 a0» y 0+a 0» ^g-ag) (26)

form a basis of P(ag, 0, yq, 6g), and the monomials

e(ag-l, 0, YQ+1. 6g), • ••> e(0, ag-1, Y Q+ag , ^g-ao+l), (27)

a basis of P(ag-i, 0, Y0+ ^> Restricted to P(ag, 0, Y0> <$o)> i-K by

virtue of (24), represented by the re

CtQ 1 0 ...

0 ag-1 2 ...

0 0 ag-2

• • « ••• •••

0 0 0

0 0 0

0 0 0 ...

^angular band matrix

0

0

0

• • •

O'" 2

2

0

an-1

0

0

0

» • •

0

0

a 0

(28)

The first ag columns of (28) constitute a square matrix whose determinant ( =

ag! ) is not zero. Hence (28) is of rank ag and of nullity 1, which means that
30

Ker P(aQ, 0, Y0> Sq) is generated by the special polynomial G(yq> <$0“a0> °*°

“0) •

In class III, the restriction of corresponds to the band matrix

a 0

1 0

0 acQ-l 2

0 0 a 0“2

• • ••• •••

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

ao-6o+2 6 Q-l

0 ao~<5 o+l

0 0

0

5 0

a 0-<5o

(29)

when the vectors

e(aQ, 0, Y0» 5 0)> •••> e( ciq-5 0 » 5 0> Y0+$0> 0) (30)

are choosen as a basis in P(ciq, 0, yq, 5q) while the vectors

e(aQ-l, 0, Y o+l > 5 0 ) » •••» e (o‘0“<5 0“ 1 » 5 0> I04l5 0+1 » (31)

are taken for the basis of P(aQ“l, 0, Y0+1> 5 q)* Matrix (29) is square, and It

is clearly non-singular. Therefore L{£ is an isomorphism of P(aQ, 0, Y0> <$q)

onto P(aQ-l, 0, Y0+1> 5 o)*

Now, in class IV, the bases in P(aQ, Bo> 0, 5 q) and P(ocq, Bq- 1> 0> <5q+1)

are chosen to be respectively
31

e(oto» $ 0 > 0, 5q), e( 0 , Bo+aQ* a 0 > Sq-cio)

and

e(aQ> Bq“1» °> Sq+D, •••» e (°» Bo+aQ-1 * a 0> 5 o-ao+l),

so that the square

S 0

a 0

0

• • •

0

0

0

which is obviously non-singular.

Finally, in class V, the monomials

e ( a 0> B 0 , 0, $o)» •••» e ( ct 0“<5 0> Bo+Bq, 5 0 » °)

matrix representing the restriction of is

0 0

B0+ 1 0

a 0-l Bq+2

0 0

0 0

0 0

Bo+aO“2 0 0

2 Bo+ao-l 0

0 1 8 0+^0

are chosen as the basis in P(aQ, 80 » 0, Bq)

,

while the monomials

e ( a 0 > Bo~l» 0, 6 o+l ) , e(ao“6 0“l» Bo+^O, 5 q+1, 0)

form the basis in P(aQ, Bo~1» 0, Bq+ 1)* In this way, the operator is

by the band matrix

(32)

(33)

(34)

(35)

(36)

given
32

B 0 0 0

a O 3 o+l 0

0 ag-l 3 0+2

• • • * * • • •

• • •••

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

ao-5o+2 3 o+<$ 0" 1 0

0 ao~6o+l 30^0

0 0 a 0-6o

(37)

The first (5g + 1) rows of (37) have the non-zero product 3 q (3+1) ••• (Bo+^o)

for determinant; therefore matrix (37) has rank 5 q + 1 and nullity 0.

That portion of the results just obtained which is needed in the proof of

Proposition 3 is summed up in the following alternative:

Lemma 4 Let (an, 3n> Yn> 3n) be the lowest quadruple in an equivalence

# class E(a, B, Y, 3). _If Bo - 0 and ag < 5 q, then Ker L^ P(a, B, Y, 3) = 0

is of dimension one , and is generated by the special polynomial

G( Yo > 3Q-ao, ao) • Otherwise Ker Lr P(a, 3, Y> 3) = 0.

Proposition 3 affords an easy way of decomposing P n into a direct sum of

irreducible weight spaces : each polynomial vq = g(a, 3, y) generates a basis

formed of the chain of polynomials v^ (k > 0) such that kv^ = L^ vk-l f° r

k > 1. Such polynomials span a weight space of weight X = a + 3 - 2y . Yet

attempts at using the complete reduction (18) to decompose any polynomial into

its constituents in Im L^ and Ker Lp have not resulted in clear and

elegant software procedures. All the same, an algorithm based on the classes

•o«

•49

enumerated in Table V proved to be both expedient and easy to code.
33

5. THE DECOMPOSITION ALGORITHM

For most dynamical systems, normalization to degree 3 is sufficient;

fortunately, at that minimal degree, the decomposition is readily executed by

hand. Once Table V has been established, it becomes clear that the polynomial

p = l C x

a

y

S Y5 (o, 0 , y , 5) a,6 ,v ,6

may be written as the sum p = p^ + q where

q = c 2, 0,1,0 x3 + - (°i, 1,1,0 + c 2,o,o,i) x2 y

1 ,1 , + “ ( c0 , 2 , l ,0 + G l , l , 0 , i ) xy z + - C 0 , 2 , 0,1 Y

6

1 o

1 1 + " c l, 0,2,0 X

^ x + - C0 , 1,2,0 xyX + - C0 , i , i , i y

2 X (38)

1 ?

1 1 , +

J

C l , 0 , 1,1 x Y + “ C 1 , 0 , 0,2 xyY + - C°» 1 » 0 > 2 y Y

+ co, 0,3,0 xx2 + Go, 0,2,1 yx2 + G0 ,o,l,2 xy2 + G 0,0,0,3 yY2

1 1 " - C0, 1,2,0 xG + - C 1 *G ,0 ,2 yG
34

and

PD * c3, 0,0,0 x3 + c 2, 1,0,0 x

2 y + C 1>2 ,0,0 xy

2 + c0,3,0,0 >

t3

2 1 2 1 + (“ C

2, 0,0,1

* - Ci, l,l,o) xG - (- C0 , 2 , 1,0

’ “ C 1,1,0,1> 7$ >

which, by virtue of Proposition 3, is an element of Ker Lp.

However trivial the task appears to be at degree 3, Table III leaves one

to gather that the calculations become rapidly voluminous past degree 4, even

for a computer program prepared to handle sparse matrices. The next

proposition is the foundation of a decomposition algorithm complete to the

point of having been coded eventually in APL^ and run on a DEC-20 at the

National Institutes of Health in Bethesda, MD. (N. B. The program is

available upon request from the third author)

.

Let V be a vector space; let also v and w be two endomorphisms of V, and

set u = vw. For any x in V, a vector y in V satisfies the relation

v(x - wy) = 0 if and only if uy = vx. Thus it is true that , given any

polynomial P Pn , £ polynomial q in ?n satisfies the relation

Lp(p-L^q)=0 if_ and only if it is a_ solution for q of_ the equation

Tq = Lp p where T is_ the operator Lp L^.

Given an endomorphism u of the vector space V, one can always find an

endomorphism u~ such that u u~ u = u. We call u~ a generalized inverse of u,

but the reader should note that authors interested in classifying various

species of generalized inverses associated with u would name u~ a

{ 1

}

-inverse 2 Q of u or a g-inverse . 2 ^ For any vector x in Im u, u u“ x = x. In

particular, assume that u is the product u = vw of two endomorphisms, and that

Im v = Im u; then any vector y of the form y = (u“ v)x is a solution of the
35

equation uy = vx. For there exists by hypothesis a vector z such that vx = uz

,

hence uy = (uu“ u)z = uz = vx. Applied to the operator T = k ~K : Pn

-* Pn

for which Im Lp = Im T (Proposition 1), the above considerations prove the

following

Lemma 5.- Let T~ _be a. generalized inverse of T and p a. polynomial in Pn

.

Then the polynomial q = T“ Lq p

^s_ £

solution of the equation Tq = Lp p.

On account of Lemma 5, after a generalized inverse T“ has been produced,

the problem of decomposing a polynomial into the sum of its constituents in

Im L^ and Ker Lq will be solved by setting Pk =

q and pp

=

P “ PK*

Among the many varieties of generalized inverses associated with T,

preference should be given here to those which preserve the basic symmetry

consisting in exchanging the coordinate x and its conjugate momentum X

respectively with the coordinate y and its conjugate momentum Y. There is

indeed no reason why the normalization should favour one coordinate more than

the other. The symmetry requirement is best expressed by introducing the

operator Z : p

•+• Zp = p(x, y, X, Y) - p(y, x, Y, X) : Pn

Pn. A polynomial p

is symmetric in the pairs (x, X) and (y, Y) if and only if Zp = 0. Furthermore

the differential operator

T = k) LK = (x

3 3 3 2

3 23 2

3 2 — + y —) + (xX + yY ) + (xY + yX ) 3x 3 y 3x3X 3y3Y 3y3X 3x3Y

being invariant for the symmetry, the operators T and Z commute over Pn . Our

purpose thus is to find a generalized inverse of T with corresponding symetry

properties. It should map symmetric polynomials onto symmetric polynomials,

that is ZT~ p should be = 0 whenever Zp = 0; also, by exchanging the paihrs
36

(x, X) and (y, Y) in the image q(x, y, X, Y) = T“ p(x, y, X, Y) , one should

have that q(y, x, Y, X) = T” p(y, x, Y, X), that is, ZT~ = T~ Z. It will be

shown that a generalized inverse called the group-inverse of T satisfies

these symmetry requirements.

For an endomorphism u of a vector space V, Erdelyi^2 calls group-inverse

of u an endomorphism u^ such that

u u^ u = u , u^ u u

^ = u^ , u u^ = u^ u .

If it exists, the group-inverse of u is unique. If u is an isomorphism of V

onto itself, then u^ = u“l; more generally, u admits a group-inverse if and

only if V may be decomposed^ into the direct sum of Ker u and Im u. This is

the case when u is semi-simple; then the restriction u# of u to Im u is

. bijective, and, in principle at least, the group-inverse may be built as

follows

0 for p in Ker u,

u’^ p

= ••©

u#" 1 p for p in Im u .

Lemma 6.- Let V be a. vector space , and u an endomorphism of V. _If u

admits a_ group-inverse u^ , then , for any endomorphism v of. V , the relation

vu = uv implies the relation vu

,if = u'

V|f v. _In particular , for any vector x of V,

the relation vx = 0 implies the relation (vu^)x = 0.

The lemma is proved when it is shown that (vu^)x = (v^ u)x first for x in

Ker u and then for x in Im u. In each case, the demonstration rests on the
37

fact that, because u and v commute, v(Ker u) is contained in Ker u and v(Im u)

in Im u.

If ux = 0, then, on the one hand, u^ x = 0 because u^ admits Ker u as its

null space22 } hence (vu^)x = 0; on the other hand, u(vx) = (uv)x = 0, which

implies that (u^ v)x = u^(vx) = 0. Therefore, when restricted to Ker u, vu^

and u^ v are identical, since they are both equal to the null endomorphism.

Now take x in Im u. There is a unique element y in Im u such that uy = x

and u^x = y. With x and y both in Im u, the elements vx and vy are also both

in Im u. But u(vy) = (uv)y = (vu)y = v(uy) = vx, and vy is therefore the

unique element of Im u mapped by u onto vx. Hence vy = u

; ^(vx), and the latter

relation implies that (vu^)x = v(u^ x) = vy = u^(vx) = (u^ v)x. This completes

the proof of Lemma 5.

The operator T = Lp is semi-simple (Proposition 1), hence it admits a

group-inverse T^. By virtue of Lemma 6, the group-inverse T^ commutes with the

symmetry operator Z, ^nd it maps symmetric polynomials onto symmetric

polynomials

.

The construction proposed here for the group-inverse T^ of T makes use

of the decomposition of ?n into a direct sum of vector subspaces P t =

P(a, £, y, 6) specified in Section 4. Assume that, for each i, the restriction

T l of T to P t admits a group-inverse; then, for each polynomial p of Pn decomposed into the sum p = £x P; of its components in the summands P t , define

the image T^ P = Tt ^ p ;

. Manifestly T^ is a linear map Pn Pn , and it

satisfies the three conditions T T

# T = T, T# T T# = T

#

, T = T# T, which

means that T^ is the group-inverse of T. The decomposition of a general
38

polynomial p into its components in Ker Lj) and Im is thereby reduced to the

problem of building the group-inverse for the restrictions of T on each of the

vector subspaces in the classes enumerated in Table V. Statistics collected in

TABLE

VI Table VI for a homogeneous polynomial of degree 6 will convince the reader

that, however tedious it may be, a careful discussion of each particular

situation breaks up the general problem of producing the 84 x 84 matrix for

the group-inverse T* into the solution of 42 linear systems in at most 3

unknowns, 22 of them being utterly trivial.

Class I is dealt with at once : T t being the null endomorphism, its

group-inverse T^ is also the null endomorphism.

In classes III - V, the factor is injective; since Ker T t is equal to

the kernel of restricted to (Proposition 1), there follows that T t is an

isomorphism of P ; onto itself, hence that T t ”^ is the group-inverse of T t . Now

a closer examination of classes III and IV will bring forth a straightforward

procedure for inverting Tf

.

As was shown in Section 4 for class III, the map is an isomorphism of

P(aQ, 0, yo> So) onto P(ao~l, 0, yo+1> <Sq) represented by matrix (29) for the

bases (30) in P(aQ, 0, yq, 6q) and (31) in P(ao~l, 0, Y0+ 1> $q) • But, for tlie

same bases, LD : P(ao- l, 0, Y0+ 1> Sq) P(aQ» 0, YQ> <$q) is represented by

the (5 q + 1) x (5 q + 1) matrix
39

Y0+1 0 0 0 0 0

5 0 YQ+2 0 0 0 0

0

t—

H

l

o Y

0"*” 2 • • • 0 0 0

• • • • • • • • • • • • • • • • • • • • • (39)

0 0 0 Y 0+<5 0-1 0 0

0 0 0 2 YO+^O 0

0 0 0 0 1 Y o+1

which is evidently non-singular; hence k> is an isomorphism of

P( a0“l , °» vo+1 > 5 o) onto P(aQ, 0, Y0» Because matrix (29) is upper

triangular, and matrix (39) lower triangular, the equation T t q

= is

solved readily first by forward substitution to obtain a polynomial r such

that bp r = p t , and then by backward substitution to find the polynomial q

such that Li£ q = r.

One meets a similar situation in class IV. The factor is an

isomorphism of P(ciq> 0q, 0, 5q) onto P(ao> 3o“l> represented by

matrix (34) when bases (32) and (33) are selected in the subspaces

P(aQ, Sq> 0, 5q) and P(ao» 3o~^» 0, 5 q+ 1) respectively. Further, for the same

bases, Lq : P(aQ» 00“^ » * P(ao> 3r)> 0 >

<5 q) is represented by the

(aQ + i) x (ccq +1) matrix
40

5 o+l

0

0

1 0

S 0 2

0 S 0-l

• • • •

0 0

0 0

0 0

5 0“ a0+3 a 0-1 0

0 5 0-ao+2 ao

0 0 6 Q-a o+

1

(40)

which is non-singular; thus Lq is an isomorphism of PCcxq, 6o~^> 0> 5(3+1) onto

P(an, 6 q» 5q). Matrix (34) being lower triangular, and matrix (40) upper

triangular, the equation T x q = p x is solved first by backward substitution

to obtain a solution of the equation Lq r = p x , and then by forward

substitution to solve the equation q = r.

The solution is not that simple in Class V. On the one hand, L^; is

injective but not surjective. On the other hand, the linear map Lq :

P ( cxq , 3q-1, Sq+1) - P(ag» 3o> 0, 6g) is surjective but not injective.

Indeed, with (35) and (36) chosen as the bases in P(ag> Bg> 0> 5 q) and

F(ag, 8o~l> 0, 6 o+l) respectively, Lq is represented by the band matrix
41

60+ 1 1

0 S 0

0

• • • • •

0 Sn-1

• • » •

• • • • •

0 0

0 0

0 0

• • • •

Sq-1

2

0

• • * ••• ••• (41)

0 0

S 0 0

1 S o+l

with ( 6 o + 1) rows and (Sq + 2) columns, and it is easily seen that matrix

(41) is of maximum rank 6 q + 1 and of nullity 1. For the restrictions of T t in

class V, there seems to be no quicker way of inverting T x than by actually

multiplying matrices (41) and (37) row by column, thereafter calculating

explicitly the inverse of their product.

In both classes I and II, the operator T t is not invertible. While, in

class I, the group-inverse of is the null endomorphism, in class II is

encountered the case where a way of computing the group-inverse T ^ must be

prescribed. But the conditions defining the group-inverse characterize u, in

the terminology of Drazin, 24 as a pseudo-invertible element in the associative

algebra End V of the endomorphisms of V. Thus the group-inverse u^ is a Drazin

pseudo-inverse of u relative to which the index of u is equal to 1. In that

context, recall the following statement, which is in fact a particular case of

a general theorem proved by Cline. 25 Let U and V be finite-dimensional vector

spaces; let also v be an injective linear map U V, and w a surjective linear

map V -* U; if the product u = vw : V -> V is pseudo-invertible in the sense of

Drazin and if its index is equal to 1, then wv : U + U is bijective, and
42

// t \-2 nir = v (wv) w (42)

The conditions under which Cline’s formula (42) may be applied are

satisfied by any endomorphism T, in class II. Indeed, from Section 4, it is

already known that LK : P(aQ, 0, Y0> 5 o) * P(ao~l» 0, Y0+1 > <$o) is a

surjective linear map; there remains to show that Lp : P(ao~l, 0, Y0+1> Sq) +

P(aQ, 0, Y0> <5q) is injective. For the bases (27) and (26) in

P(ao~l, 0, Y0+ 1> <5q) an<^ ^( a 0> 0, Y0> <5q) respectively, Lp is represented by

the band matrix

Y o+ 10 0

6 o Y 0+2 0

0 ’ 5 q— 1 Y

0

+3

0 0 0

0 00

0 0 0

0 0 0

0 0 0

0 0 0

6 0-ao+3 yo+^O-1 0

0 6()-ao+2 Y 0“a 0

0 0 5 q—a q+ 1

(43)

with ap+1 rows and ctQ columns. Clearly, the determinant made of the first ao

rows of (43) is not zero; hence Lp restricted to P(aQ”l, 0, Y0+ ^> Sq) is

injective. In application of formula (42), the computer program computes the

group-inverse
43

T \

//f = = LD (Lr Lp )“2 Lk

by multiplying matrix (28) by matrix (43), squaring the product, taking its

inverse, and multiplying the result to the left by matrix (43) and to the

right by matrix (28). Considering that the matrices involved have dimensions

of the order of half the degree n, one will admit that the manipulations have

been brought down to an elementary level. Furthermore, by taking advantage of

the invariance with respect to the symmetry Z, the task of constructing the

group-inverse has been cut by almost a half. Alternatively, the invariance may

be exploited to check the results coming out of the program.

The fundamental results obtained in the course of discussing the

restriction classes mentioned in Table IV are gathered in the following

Proposition 4.- Given sl polynomial p in ?n.‘ there exists a_ unique

polynomial q in Im Lp such that p - Lr q belongs to Ker Lp. Also Zq

belongs to Im Lp, and it is the unique polynomial r in Im Lp such that

Zp - I,£ r belongs to Ker Lp. Whenever Zp = 0, then Zq = 0.

Indeed p may be decomposed in a unique way as the sum P = Pr + Pd with p^

in Im Lr and pp in Ker Lp. Hence there exists a polynomial q such that Lp p =

Lp Lr q. Taking q = T^ Lp p guarantees that q belongs to Im T which, by virtue

of Proposition 1, is identical to Im Lp. If there is another polynomial q'

with the same property,, then, on the one hand, q - q* belongs to Im Lp, while,

on the other hand, q - q* belongs to Ker T = Ker Lj^; but Pn

is the direct sum

of Ker Lr and Im Lp, hence q - q

T = 0. In view of the fact that the

expressions (9) and (12) for Lr and Lp are symmetric in the pairs (x, X) and

(y , Y) , Lp Z = Z Lp and L^ Z = Z L^; therefore Zq belongs to Im Lp and
44

Lq (Zp - Zq) = Ld (Z(p - Lr q)) = Z(Ld (p - q)) = 0. From what has

been proved already, there results that Zq is the unique element in Im Lp to

have that property. Finally, the last part of Proposition 4 is an immediate

consequence of Lemma 6.

Incidentally the computer program which implements the algorithm produced

at degree 3 a polynomial q = Lq p which differs from the solution in (38)

by the quantity

2~ G < c 0 , 0 , 2,1 x " c0 , 0 , l ,2 Y )

The discrepancy is admissible since, in agreement with Prposition 1 , it is an

element of Ker L^.
45

6. ELIMINATION OF THE SHORT TERM EFFECTS

A dynamical system described by an Hamiltonian of type (14) is said to be

in normal form if, for each n > 2, the term Hn belongs to Ker Lr>, that is, Hn

is a homogeneous polynomial in the coordinates (x, y) and the angular

momentum G (Proposition 3). For instance, an Hamiltonian whose potential

energy depends only on the coordinates is in normal form; so there is nothing

the present normalization can contribute to further its solution. Such is the

case for the Monkey Saddle . 26 But, if an Hamiltonian of type (14) is not in

normal form, then one can construct a canonical transformation

(x, y, X, Y) * (x* , y* , X', Y') which will convert (14) into a series in

normal form. It is proposed to construct the normalization as a Lie

transformation, that is, as the flow of an Hamiltonian vector field

dx aw dy aw dx aw dY aw

de 3X de 3Y de 3x de 3y

derived from a series

„

1 w = w(x, y, x, Y) = I — w

*n+1 . n>0 n •

It will be shown that each term Wn

in the generator may be obtained as a

homogeneous polynomial of degree tri-2 in the phase variables (x, y, X, Y) , so

that the normalization may be pursued in a recursive fashion from one degree

to the next
46

Starting with the infinitesimal contact transformation that is the

infinitesimal symplectic transformation tangent to the Lie mapping, the first

order terms H’ j and Wj in the new Hamiltonian and in the generator

respectively are linked by the identity

With the requirement that Lp H'x = 0, the problem of solving (44) amounts to

decomposing the homogeneous polynomial into its constituents in the direct

A change in notation is helpful in following the recursion rules for

constructing the normalization past the infinitesimal contact transformation:

r*n,0 will stand for Hn in (14), and HQ >n for H' n

. Now assume that all terms

(1 < i < n-1) and ^i,j (0 < i, j < n-1, i+j S n-1) have been obtained. Then

one is in a position to calculate first

(K; W L

) + Hi = H»

i

equivalent, as a matter of fact, to the algebraic relation

Lk WL + H * i = Hi (44)

sum P 3 = Ker Lp Im . •00

^n- 1,1 “ ^n,0 + I

n-1

( ) (

H

n_

j

} 0 > ) l<j<n-l j-1

and thereafter, by decreasing values of i, the terms

0<k<i k
47

for i + j * n, 0 < i, j < n, the last term in the recursive chain being Ho jU .

By prescription of a perturbation algorithm involving a Lie transformation,

the terms Wn

in the generator and Ho >n in the transformed Hamiltonian must

satisfy the partial differential identity

(Ho; Wn ) + H0 , n - H0 , n • (45)

But, if p and q are homogeneous polynomials of degree i and m respectively,

then their Poisson bracket

3p 3q 3p3q 3p3q 3p3q

(p; q) +

3x 3

X

3X 3x 3y 3Y 3Y 3y

is a homogeneous polynomial of degree l + m -2. Hence* all terms ^i,j» f° r

i + j = n, are homogeneous polynomials of degree n + 2, and the partial

differential equation (45) is equivalent to the algebraic equation

Tti0,n = Wn + Ho, n •

With the normalization requirement that Lp Hq )T1 = 0, the latter is solved by

decomposing Hq >h into its constituents in the direct sum Pq+2 =

Im Lk Ker Lq.

For readers interested either in calculating by hand some of the

coefficients in the normalized Hamiltonian, or in checking their automated

normalization procedures, listed below are the terms in HQ >n (3<n<6)

emanating from the right hand members

•••
J 48

HO, n “

(a,

7 C x’ a y'S X'T Y' 5: 3, Y, 5) a,B,y,5

Degree 4

h0,4 = 04 , 0 , 0,0 x

’ 4 + -

,

1 + c 3, 1,0,0 X ’V + ~

9 9

1 + 02,2,0,0 x

,2 y’ 2 - -

,

1 + c i ,3,0,0 x 'y’ 3 + T

0

+ co , 4 ,0 ,0 y’ 4

;

(3 03,0,0,1

( C2 , 1 , 0,1

(3 c0 , 3, 1,0

(2 c

2 ,0 ,0 ,2

c 2,l,l,0) x

' 2g ’

c l,2,l,0) x’y’G'

c i,2,o,i) y

,2 o’

c l, 1,1,1 + 2 Cq,2,2,o) O’ 2

Degree 5

C- 1 H

0 , 5 = c 5, 0,0,0 x + - (4 C4,o,0,l “ 03 , 3 , 1 , 0 ) X

’ 3 G'

+ 04,1,0,0 X? V + - (3 c 3 , 1,0,1 - 2 c 2 ,2,i,o) x

' 2 yo f

+ 03,2,0,0 x

* 3 y

?2 - “ (3 Cl, 3, 1,0 - 2 c2 ,2 ,0 , 1 ) x ’y ,2 G'

* J

+ c

2 , 3 ,0 ,0 x

’ 2 y' 3 - - (4 c0 , 4, 1,0 - c i ,3 ,0 , 1) y

,3 G

f

+ c i , 4 ,0 ,0 x 'y' 4 + 7

(3 03,0,0,2 - 0 2 , 1,1,1 + Cl, 2, 2,0) X ’G’ 2 O

, 1 + co , 5 ,0 ,0 y’ 5 + 7

(3 Co, 3, 2,0 “ C 1 ,2 , 1

, 1 + 02,1,0,2) y'G' 2

; o
49

degree 6

h0,6 = c6, 0,0,0 x ’* + ~ ( 5 c 5, 0,0,1 - c4»l,l,o) x' 9 G’

c

5, 1 , 0,0 x '}y' + ~

( 2 c4, 1,0,1 “ 2 c 3f2 ,i,o) X, VG

’

+ c

4 ,2 ,0 ,0 x

' 9 y* 7 + - ( c

3 ,2 ,0 , 1 - c2,3,l,0) x'V 7 G’

+ c

3 ,3,0,0 x

,8 y» 8 -

j

( 2 C 1>4>1>0 - 2 C2>3>0 ,1) x'y’ 8 G’

+ c 2,4,0,0 X 'V 9 - - ( 5 C0)5)1>0 “ C

1 ,4 ,0 , 1 > y' 9G ’ 0

+ C 1 ,5

,

0,0 x 'y'> + “ (12 G4,0,0,2 - 3 c 3, 1 , 1,1 + 2 C 2j2>2>0 ) x

’

G

1 . + c

0 , 6 ,0 ,0 y

?+ + “ ( 3 G 3 , 1 ,0 ,2 ~ 2 C 2>2jl>1 + 3 C 1>3j2>0 ) x'y 'G'

»7 P ,7

1

« + “ 7 ^i7 (12 G

0

,

4 ,2 ,0 - 3 Ci, 3, 1,1 + 2 C 2)2>0 ,2) y

f ‘ G

1 + ” ( 3 G

3 ,0 ,0 ,3 - G 2 , 1

, 1 ,2 + Ci, 2, 2,1 “ 3 c0,3,3,0>

-

1

8

Let the relativistic corrections for a free particle illustrate the

normalization. The kinetic energy

7 7 7 — 1/2

E = ihq c (1-v/c)

expanded in powers of v

7

, after division by the mass at rest mQ and omission

of the constant energy at rest me7

, gives rise to the Hamiltonian

1 3 v

9 5 v*

_ v — — —— + —— — . . . 2 8 c

7 16 c

9

i
50

with v

2 = X2 + Y2

. Since

Lk

(xX + yY) v

2 = v

4

,

there will be no term of degree 4 in the normalized Hamiltonian, and the

generator of the infinitesimal contact transformation will be

3 v

2 Wi (xX+yY) . « n*-

Hence, by definition of an infinitesimal contact transformation

= x’ + (x T ; Wj) , X = X’ + (X T ; Wi),

1

—

+ >,

11

»—

H

><

II >-< + >-< S

the relativistic corrections to the first order in v

2 /c2 are:

Ax’t __= _

3 v * 3 G’ - — x’ + - -- V , 8 c^ 4c2 AX' =3 v

' 2 - -7- X’ , 8 c

2

Ay T =_ __ 3 v

2 3 G’ - v

* - - — y’

8 c

2 7

4 c

2 ’ AY’ =

-5 ,2

3 v - — Y’ . 8 c

2

A more substantial application of the present scheme for normalization

has been made to the Restricted Problem of Three Bodies at the equilateral

point L4 for Routh’s critical mass ratio; but this topic requires too much

background information in celestial mechanics to be related here.
51

Unexpected as it comes after a long excursion in Representation Theory

of Lie algebras, the physical meaning of the normalization achieved in the

present section is in fact very simple. To elucidate this point, the notations

will be revised. First the normalized Hamiltonian will be decomposed in the

usual way as the sum

of a kinetic energy and of a force function U, with G' = x* Y' - y* X'

designating the angular momentum. Next, in order to eliminate ambiguities

concerning the partial derivatives, the differential of U will be written as

the 1 -form

Consider now a Cartesian frame of reference consisting of an orthonorraal basis

(i', I', k) rotating at the angular velocity w = 3 3 U k about the normal to the

plane of motion; assume also that (x*

,

y’) represent the Cartesian coordinates

of the particle in the plane ( i

* , ^ ’ )

,

or that x=x' i’ + y* j

f . Under these

+ V

2

) - U(x f , y

f , G')

2

dU = 3 ^U dx' + 3 2 U dy f + 83 d dG' .

In these notations, the equations of normalized motions become

. 3H

x

* = = X’ + y' 3oU , 3Xf X’

8 H

T — - 3

1 U + Y ’ 3 3

tJ

. 3x T

3 Y*

T " Y* - x’ 3 3 U ,

3 H

? = 3 2 U - X’ 3 3 U . 3y»
52

conditons, the particle's velocity in a frame (C) fixed in the plane is the

vector

x = x' i' + y* ' + (ii x x ,

equal, by virtue of the normalized equations of motion, to the sum

x = X' i' + Y’ .

Thus the conjugate momenta X' and Y' are the components in the moving frame of

the particle's velocity with respect to the fixed frame. For the same reason,

x = X' i' + Y' ]' +u x x = ajU i' + 3 2 U i' ,

which exhibits 3 and 3 2 U as the projections of the force on the axes of the

rotating frame. The normalization appears now as a procedure to extract the

rate at which the frame of reference should be rotated in order to confer the

equations of motion the simple form

x = 3

X U i' + 3 2 U I'

.

From that standpoint, the normalization is closely analogous to a method

devised by Hansen*- 7 for handling perturbed Keplerian systems in three

dimensions. A slow rate of rotation is imparted to the frame of reference; its

axis and its rate are adjusted at each instant so that the rotating frame

constitutes what Hansen calls an ideal (i.e. conceptual or virtual) reference
53

system* In the ideal frame, the particle’s motion appears to be planar; the

forces acting on the mass point are expressed as a two-dimensional gradient,

the coupling between the planar motion and the rotation of the ideal frame

being accounted for by making the force function dependent explicitly on the

angular momentum.

The kinematical interpretation given here to the normalization is further

clarified by looking at the equations of motion in the polar variables defined

by the canonical extension

0

' x’= r’cos 9', X’= R’cos 9’- - sin 9’,

r’

0 ' y’= r’sin 9’, Y’= R’sin 9’+ - cos 9’.

r'

There results at once from the the Cartesian equations of motion that

d d 0

2

dt

11 p

dt r

* ^ + 3iU cos 9

’ + 3 2 U sin 9 ' ,

d 0’ d

dt

9’= — - 3 3U , r’ — 0’= r ’ ( dt

3 2 U cos 9 ’-

3 ]_U sin 9 ’ )

.

The angle a of slow rotation being determined by the 1-form da = 3 3 U dt, and

the radial and transversal components of the force being

P = 3 jU cos 9 ' + 3 2 U sin 9

' ,

0 = 3 2 U cos 9’ - sin 9' ,

the equations in polar coordinates are given the standard form
54

2

d d

0' — (9 » + a ) » dt r

i 2

d-- 0’ = Q . dt

Thanks to the normalization, the particle appears to move under the exclusive

action of the gradient of U with respect to the normalizing coordinates.
55

APPENDIX : CONSTRUCTION OF A GROUP-INVERSE

It has been shown in Section 5 that the decomposition of a polynomial

p of P n into its components in Ker Lp and Im reduces to the construction

of the group-inverse T^ of the semisimple operator LpLjr. The algorithm given

in Section 5 for constructing T^ depends on special properties enjoyed by the

action of T on Pn : ?n can be written as the direct sum of the spaces

P(a,S,Y,<5), each of which is left invariant by T, and on each subspace

P(a,0,y,5) either T is invertible, or the group-inverse of the restriction of

T can be computed via an explicit factorization of its matrix. The

decomposition scheme adopted in Section 5 owes its effectiveness to the fact

that the subspaces P(a,0,Y,<5) are much smaller than Pn : the dimension of P n n + 3 ~ is ( ), hence of order n-5 , while the dimension of P(a,8,Y»5) is at most

3 n/2 + 1 (by the discussion preceding Lemna 3). The purpose of this Appendix

is to sketch an alternate algorithm for constructing the group-inverse of T.

This algorithm hinges on the fact, to be proved next, that T is a

diagonalizable operator whose eigenvalues are simple to find; it does not make

use of any other properties of T or of Pn

.

Lemma. T _is_ a_ diagonalizable operator whose eigenvalues consist of the

products (n - 2y —i ) ( 1 + 1), with 0 < y < [n/2] and 0 < i * n -2y

The

number of distinct eigenvalues is less than 1 + [(n + l)/2][(n + 3 ) / 2 ] / 2 <

(n + 2

)

2

/ 7 .
56

Proof . Pn may be written, as in (18), as the direct sum of certain

subspaces (1 < k < s) , each of which is invariant and irreducible under the

set of operators (Lq, L^, Lg} . Furthermore, in each V^, there is a basis

(k)

0

(k) v0

“ v„ > • • » > vVmm

= v satisfaying relations (19) - (21) (with x, y, h

replaced by Lq, L^, Lg respectively). It follows from (20) and (21) tha the

Vj_'s (0 < i < m^) are a basis of eigenvectors for T = LpL^, with Tv^ =

(m^ - i)(i + 1) vj_. Thus the eigenvalues of T on consist precisely of the

products (m^ - i)(i + 1) (for 0 < i < m^) . To see which m^'s arise, one may

(k)

observe, on the one hand, that in each the maximal weight vector v is an

eigenvector of Lg of eigenvalue m^. Furthermore, since the s linearly

(k)

independent vectors v^ (1 < k < s) form a basis of Ker Lq

,

(by relation (21)

and Proposition 1), the set (m^ : 1 < k < s} consists of the eigenvalues which

Lg takes on Ker Lq. On the other hand, the nonoraials g(a,3,y) = x

a y$ Gl with

a + 6 + 2y = n are, by Proposition 3, also a basis of Ker Lq, and are

eigenvectors of Lg as the following formula shows:

kg g(a,8,Y) = (a + S) g(a,3,y) = (n - 2y ) g(a,3,y).

Thus the numbers m^ ( 1 < k <, s) are exactly the numbers n - 2y (0 < y < [n/ 2 ] ) •

Finally, for a fxed y, there are at most [(n - 2y + 1 ) / 2 ] = [(n + 1 ) / 2 ] - y

distinct non-zero products among the numbers (n - 2y - i)(i + 1) (for

0 < i < n -2y), so, in all, T has at most 1 + T [ (n + 1 ) / 2 ] - y = CKy< (n/2)

1 + [(n + l)/2][(n + 3 ) / 2 ] /

2

distinct eigenvalues. It is easy to check that

2 this number is always smaller than (n +2) /7.

A glance at Table VI indicates that the bound given in the Lemma is

far too generous.

It is perhaps worthwhile to note that a slight extension of the argument
57

just presented yields a description of the decomposition of Pn into its

irreducible components : for each y satisfying 0 < y < [n/2], the irreducible

sl(2, R)-module of highest weight n - 2y occurs with the multiplicity

n - 2y + 1 , and no other occur. This can be derived without difficulty from

the fact that the polynomials g(a,B,y) generate irreducible subspaces of

highest weight n - 2y .

In order to state the main result of the Appendix, it will be convenient

to have some more notation. If X is a scalar, let X^ denotes X~^ if X * 0,

and 0 otherwise. If Xq, ..., X t is a sequence of distinct scalars, then there

is a unique polynomial f(x) of degree < t satisfying f(X k) = X ^ for 0 < k < t.

k

If f[XQ, ..., X k ] denotes the k-th divided difference (constructed, for

example, from the prescription f [

X

q

]

= f (X q) and

f[Xi, ..., X k ] — f[XQ, . . . , X k— 2

]

f[XQ> •••> X k ] = —————————————————— —

x k

" ^0

for k > 1), then, as is well-known, f(x) can be written as

f ( x ) = + 1 f [Xq » • • • >Xk) (x - X k_i)***(x - Xq).

1 < k< t

A number of authors^ have observed that the group-inverse T

1^ can be obtained

as a polynomial in T. In the proof of the next proposition, it will be seen

that, for the case considered here,

T# = f(T) = f [X q] 1+7 f[X 0 ,...,X k ] (T - Xfc-i ,1) • • *(T - X 0 I).

l<k< t
58

Proposition. Let T _be £

diagonalizable operator on £ finite-dimensional

vector space , and suppose the distinct eigenvalues of T are Xq, . . . , X^ . For

any vector v in V, w = v can be computed by the following algorithm:

Set v0 = v, w0 = f[X 0 ] v0 ;

For k = 1 to t ,

set v^ = (T - X k_i) vk_!, wj*. = wk_ 1 + f[X 0 ,...,X k ] vk ;

Then w = wt

.

[ N. B. The cycle in k may be stopped as soon as it emcounters

a k for which vk is = 0. ]

Proof . It is clear that wkis just the k-th partial sum of the sum

f(T) v = f [ X o 1 v + I f [Xo> • • • >X k ] (T - X k_i I) ••• (T - Xq ) v,

l<k< t

so wt = f(T) v. It only remains to see that f(T) v = T^v for all v in V.

Since T is diagonalizable, V may be written as the direct sum Vq + • •

• + Vt of

the eigenspaces of T, where Vk = {v V : Tv = X k

v} for 0 < k < t. But the

endomorphism of V which sends u = uq + • •

• + ut (uk

Vk

for each k) to Xq^

uq + ••• + x t # u t satisfies the relations for the group-inverse of T, since Tu

= Xg uq + • •

• + X t u t

. Thus by uniqueness of the group-inverse, T^u =

Xq^ uq + ••• + X t

,,f ut

. On the other hand, f(T)u = yf(T)u t , and the relation

Tuk

= ^k uk implies easily that f(T)uk = X k

'^ uk

for each k.

f(T )u = y X k

^ uk = T^ u for all u in V, and the proof is complete.

Hence
59

It is worthwhile to place this algorithm in a somewhat wider context. For,

clearly, the method just outlined may be used to compute any polynomial in T,

not just the specific polynomial which yields the group-inverse. The range of

applicability of this algorithm is thus determined exactly by the following

theorem in linear algebra^^

Theorem. Let S and T be linear operators in a finite-dimensional vector V V V \ V S V ' - — - I — II — I , — — — •- - — . space V. Then S may be written as a. polynomial in T if_ and only if S commutes

with every linear operator which commutes with T, that is , if and only if for

every linear operator L in V, the relation LT = TL implies LS = SL.

If S and T are diagonalizable , the theorem is quite easy to prove

directly. The special case when T is diagonalizable and S = T^ has been

confirmed in the Proposition of the Appendix and in Lemma 6.
60

a0n leave from the University of Rochester, Rochester, NY 14627, U.S.A.

*E. T. Whittaker, Proc. London Hath. Soc. 34, 206-221 (1902).

^C. Delaunay, Theorie du mouvement de la lune . Memoires de I'Institut

imperial des Sciences (Paris 1863 and 1867).

^E . T. Whittaker, Proc. Royal Soc. Edinburgh 37, 95-116 (1916). For an

automated development of the generating function, see e.g. F. G. Gustavson,

Astron. J. 71, 670 - 686 (1966).

^H. Poincare, Bull, astron. 3, 57-61 (1886); G. R. Acad. Sc. Paris

108, 21-24 (1889); Les methodes nouvelles de la mecanique celeste , Vol. 2

(Gauth.ier-Villars , Paris 1893).

^A. Deprit, J. Henrard, and A. Rom, Icarus 6, 381-406 (1967).

^A. Deprit, Cel. Mech. 1, 12-21 (1969); A. H. Nayfeh, Perturbation

Methods (John Wiley and Sons, New York Chichester Brisbane Toronto, 1973);

J. R. Cary, Phys. Reports 79, 131 - 159 (1981).

^E. T. Whittaker, A. Treatise on the Analytical Dynamics of Particles and

Rigid Bodies (Cambridge University Press 1937).

8j. Williamson, Am. J. Math. 58, 141 - 163 (1936).

9j. - C. van der Meer, : Mathematisch Instituut der Rijksuniversiteit

Utrecht Preprint Nr 169, 1980 (accepted for publication in Cel. Mech.)
61

10m. Kummer, Commun. Math. Phys. 48, 53 - 79 (1976).

1

1 A • Deprit, "Methodes nouvelles de simplf ication en mecanique celeste”.

Publications du Departement de Mathematique , Facultes universitaires Notre

Dame de la Paix, Namur, Belgium, 1980 .

12r. Cushman, and D. Rod," Reduction of the Semi-simple 1:1 Resonance.”

Preprint of the Departement of Mathematics and Statistics, University of

Calgary, 1981 .

l^A. Deprit, Cel. Mech. 26^ 9-21 (1982); "The Reduction to the

Rotation for planar perturbed Keplerian Problems", 1982 (accepted for

publication in Cel. Mech.).

l^See e. g. R. Cushman, Symposia Mathematics 14, 323 - 342 (1974)

l^E. Cartan, Lecons sur la theorie des invariants integraux (Hermann,

Paris, 1922).

l^The theorem was first stated by V. V. Morozov in the C. R. (Doklady)

Acad. Sc. URSS 36, 83-86 (1942), but proved later on by N. Jacobson in the

Proc. Amer. Math. Soc. 2, 105-137 (1951), finally restated and proved again by

V. V. Morozov in the Uspek.. Mat. Nauk 15, 137 - 139 (1960).

^Arnold, V., Mathematical Methods of Classical Mechanics (Springer, New

York Heidelberg Berlin, 1978), p. 211.
62

18j. E. Humphreys, Introduction to Lie Algebras and Representation

Theory , Third printing, Revised (Springer, New York Heidelberg Berlin, 1980).

l^K. E. Iverson, A programming language (John Wiley and Sons, New York,

1962) .

-^A. Ben - Israel, and T. E. N. Greville, Generalized Inverses : Theory

and Applications (John Wiley and Sons, New York - London - Toronto - Sidney,

1974)

.

21c. R. Rao, Linear Statistical Inference and Its Applications , 2d

Edition (John Wiley and Sons, New York, London, Toronto, Sidney, 1973).

22 I. Erdelyi, J. Math. Anal. Appl. 17, 119 - 132 (1967).

22 P. Robert, J. Math. Anal. Appl. 22, 658 - 669 (1968).

2 ^P. Drazin, Amer. Math. Monthly 65, 506 - 513 (1958).

2 ^R. E. Cline, "An application of representations for the generalized

inverse of a matrix." Technical Report 592, U.S. Army Mathematics Research

Center, University of Wisconsin - Madison 1965.

26D. Rod, J. Diff. Equat. 14, 129 - 170 (1973).
63

27p # A. Hansen, Fundamenta nova Invest igationis Orbitae verae quam Luna

perlustrat (Glaser, Gotha, 1838); E. W. Brown, An Introductory Treatise on the

Lunar Theory (Cambridge University Press 1896); H. Andoyer, Cours de mecanique

celeste (Gauthier-Villars , Paris, 1923); A. Deprit, J. Res. Nat. Bur. Stand.

Sec. B, 79 1 - 15 (1975).

2&S. Campbell, and C. Meyer, Generalized Inverses of Linear

Transformations (Pitman, London San Francisco Melbourne, 1979), pp. 130 - 136.

2%. Jacobson, Lectures in Abstract Algebra , Vol. II, Linear Algebra (D.

Van Nostrand Co. Inc., Princeton, Toronto, London, New York, 1953), p. 113.
64

TABLE I. A basis in the algebra sp(4, R)

Generator

Type

Non zero elements

n=nilpotent

s=semi-simple

wi - (X2 + Y

2

) 2 W1 ,1,3 = w i ,2 ,4 = 1 n

W2 xX + yY w2 ,1,1 = w2 , 2 , 2 = _w2 ,3,3 = "w2,4,4 = 1 s

W3 xY - yX w3 ,2,1 = w3 ,4 ,3 = ~w3 ,1,2 = ~w3,3,4 = 1 s

w4

j

(X2 - Y

2

) w4 , 1 , 3 = “w4 ,2,4 = 1 n

w5 XY w5 , 1 , 4 - w5

,

2 , 3 = 1 n

wg - - (x2 + y

2

) w6 , 3

,

1 “ w6 , 4

,

2 = 1 n

w7 xX - yY w7 ,1,1 - ~w7 ,2,2 = _w7 ,3,3 = w7,4,4 * 1 s

wg xY + yX w8 , 1

, 2 = w8 , 2 , 1 = “w8 ,3,4 = ~w8,4,3 = 1 s

wg - xy w9 , 3 , 2 = w9 , 4 , 1 = 1 n

1

, , w10 - - (xz - y

z

) w10 ,3 ,

1

= ~w10 ,4 ,2 = 1 n
65

TABLE II. Commutators of the basis matrices in sp( 4 , R)

W1 W2 w3 w4 w5 w6 wy w8 wg w10

W1 0 -2w] 0 0 0 W2 -2w4 -2w5 w8 wy

2w^ 0 0 2W4 2W5 -2w6 0 0 -2wg -2wio

W3 0 0 0 2W5 -2w4 0 2wg -2w7 -2wiq 2wg

w4 0 -2W4 -2w3 0 0 w-j -2wi 0 -w3 w3

w5 0 _2w5 2W4 0 0 w8 0 -2w^ W2 w3

w6 -W2 2w6

‘ 0 -w7 -w8 0 2w 10 2W9 0 0

w7 2W4 0 -2wg 2w^ 0 -2wio 0 -2w3 0 -2w6

w8 2W5 0 2w7 0 2w^ -2wg 2w3 0 -2w6 0

wg “w8 2wg 2wiq w3 -w2 0 0 2w6 0 0

w10 -w7 2wiq -2wg -W2 -W3 0 2w6 0 0 0
66

TABLE III . Algebraic simplification accomplished by normalization

Degree 3 4 5 6 7 8 • . •

dim P n

20 35 56 84 120 165 ...

dim Ker Lq 6 9 12 16 20 25 ...
TABLE IV . Classes of restricted to P(a, 0, y, 5)

Class P(a, 0 , y , 5

)

Matrix L^ Matrix Lq

Lowest element Dimensions Rank Nullity Dimensions

I 0q = 0, ao = 0 0 1

II 0q = 0, 1 < ao < 6 q ao x (ao* 1 ) a 0

1 (ao+1) x aQ

III ' 00 = 0, ao > S 0 ( 6

0

+D x (5 o+l) <5 0+1 0 (6 o+l) x (6 o+l)

IV 30 * 0» YO = a 0

< 5 0 (ao+1) x (ao+1) ao+1 0 (ao+1) x (ao+D

V ^0 * 0, yo = 0, ao > <5 q (6 o+2) x (6 o+l) 6 0+1 0 (6 o+D x (5 0+2)
«

c—•

it

-

itit

-1

itit68

TABLE V . Image and co- image of in the space P3

v - x

2 = x

2 X

3

Lk - x

2 X = xX2 2 Lk

xX2 = X

3

1 o

l

v- - x-y = xyX + - xG

3 3

1 1 Lk ( - xyX - - xG ) = yX2 2 2 Lk yX2 = X

2 Y

r. 3

xy2 ^

y

2 X

2 1 ? - yG Lk - x

2 Y = xXY

3

~ 2 Lk xY2 = XY3

k t xZ y

= x^Y -

2 1 5 - xG LK - y

2 x = yXY

3 ~ 2

l

2 — v3 k yYz = y

I , 1 - Xy

z = xyY - - yG

3 3

1 1 Lk ( - xyY + - yG ) = xY2

" 2 2

1 - o K - y

z = y

2 y Lk y2y = yY2

2
69

TABLE VI . Count of matrix inversions sufficient

to calculate the group-inverse at degree 6.

Classes

Dimensions

lxl 2x2 3x3

II or V 5 3 1

III or IV 6 4 2

U.S. OEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR 2 REPORT NO.

NBSIR 82-2541 (NBS)

. Performing Organ. Report NoJ 3. Publication Date

June, 1982

4. TITLE AND SUBTITLE

Normal Form and Representation Theory

5. AUTHOR(S)

R. Cushman, A. Deprit and R. Mosak

6. PERFORMING ORGANIZATION (If joint or other than NBS. see instructions) 7. Contract/Grant No. national bureau of standards

DEPARTMENT OF COMMERCE 8. Type of Report & Period Covered

WASHINGTON, D.C. 20234

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10.

SUPPLEMENTARY NOTES

~| Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11.

ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here)

Representation theory of Lie algebras is called upon to develop a procedure for

normalizing a dynamical system with two degrees of freedom in the neighbourhood

of an equilibrium when the Hamiltonian H(x,y,X,Y) in the coordinates (x,y) and

their conjugate momenta (X,Y) is of the type H = (X2 + Y 2 )/2 + V(x,y,X,Y), the

potential energy V being a sum of homogeneous polynomials in the phase variables

of degree strictly greater than two. The fact that the resulting potential V' is a polynomial in the new coordinates (x',y ! ) and the angular momentum G

1 =

x

1 Y1 - y

1 X

1 implies that the normalization is a rotation in the configuration

space from a fixed frame to an ideal frame. The technique is intended for normalization is a rotation in the configuration space from a fixed frame to an ideal

frame. The technique is intended for normalizing an Hamiltonian in equilibrium

at the. origin when the Lie derivative associated with the quadratic part is not

semi-simple, e.g. the planar Restricted Problem of Three Bodies at the equilateral

equilibrium L 4 when the basic frequencies are equal (Routh's singular case).

12.

KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolon s) Generalized inverses; Hamiltonian Mechanics; Lie algebras; Non-linear oscillations;

Normalization; Representation theory

13.

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