Integrability and reduction of normalized perturbed Keplerian systems

Integrability and reduction of normalized perturbed Keplerian

systems

Citation for published version (APA):

Meer, van der, J. C. (1988). Integrability and reduction of normalized perturbed Keplerian systems. (RANA : reports on applied and numerical analysis; Vol. 8815). Technische Universiteit Eindhoven.

Document status and date: Published: 01/01/1988

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RANA 88-15 October 1988

INTEGRABILITY AND REDUCTION OF NORMALIZED PERTURBED

KEPLERIAN SYSTEMS

by

J.e.

van der Meer

Reports on Applied and Numerical Analysis

Department of Mathematics and Computing Science Eindhoven University of Technology

P.O. Box 513 5600 MB Eindhoven The Netherlands

perturbed Keplerian systems

by

J.C. van der Meer

ABSTRACT

In this paper it is shown that under certain conditions integrable fonnal nonnal fonns can be obtained for perturbed 3-dimensional Keplerian systems. The two fonnal integrals allow us to reduce the obtained integrable approximation to a one degree of freedom system and analyze its qualitative behavior. As an exam-ple the lunar problem is considered.

Key words & phrases :

Integrability, constrained nonnal fonn, equivariant nonnalization, reduced phase space, per-turbed Keplerian system, lunar problem.

Table of Contents

O. Introduction

1. Preliminaries - a first nonnalization and reduction

2. Equivariant nonnalization - nonnalization on the orbit space 3. Equivariant normalization - averaging over tori

4. Equivariant normalization - the case of polynomial coefficients 5. Reduction to one degree of freedom

6. The lunar problem - normal form

7. The lunar problem - analysis of the integrable approximation References 3 5 8 10 12 14 18 20 23

§ O. Introduction

In this paper we develop an algorithm for further normalization of normal forms for per-turbed Keplerian systems. This process called equivariant normalization respects the symmetries obtained by earlier normalization. Our approach is similar to ideas in [3] where further normali-zation of Hamiltonian systems near equilibrium points is discussed. In this paper we consider for-mal power series perturbations of Keplerian systems. Under certain conditions on the lower order terms of the perturbation further normalization is possible. When considering 3-dimensional per-turbed Keplerian systems an integrable normal fonn is obtained after normalizing twice.

A first step towards the normalization of perturbed Keplerian systems was made in [1] where it is shown that Hamiltonian systems on lR 2n with formal power series Hamiltonian H=H

_o

+EH1+E2H2+ ... ,H"E

coo

(lR 2n ), can be normalized if the Hamiltonian vector

field XHo corresponding to the zeroeth order term H 0 has periodic flow. The normalization comes

down to averaging over the periodic solutions of XHo' The resulting normal form up to order m is

-

-

-H

=

H 0

+

E HI

+ ...

,L_H 0 H"

=

{H" , H o}

=

0, 0 $ k

=

m, where {.,.} is the standard Poisson

bracket in lR2n. We speak of a normal form with respect to H o.

A second step was made in [4] where the algorithm of [1] is adjusted in order to be able to normalize Hamiltonian systems which are constrained to some symplectic submanifold of lR 2n • In [4] it is also shown that regularized perturbed Keplerian systems can be considered within the framework of constrained systems to which this constrained normalization algorithm applies.

The third and final step is further normalization of the obtained constrained normal form. This comes, mutatis mutandis, down to just applying the constrained normalization algorithm again to the obtained normal form. Let H be in constrained normal form up to order m with respect to H o. Under certain conditions on HI, H can now be normalized with respect to HI'

A

The formal power series Hamiltonian H obtained after double normalization up to order m, now commutes, up to order m, with HI as well as with H 0 because the second normalization does not

interfere with the earlier obtained symmetry with respect to H o. Consequently, when H 0

corresponds to a perturbed 3-dimensional Keplerian system, we have after truncation at order m ~ 2 an integrable system.

The integrable approximation obtained after truncation can be analyzed by reduction to a one degree of freedom system. Hereto we apply the reduction process twice, first with respect to the XHo -flow, and second with respect to the Xii, -flow. All the possible two dimensional phase spaces which one can obtain this way are described.

The contents of the paper is as follows. After some preliminaries in Section 1 equivariant normalization is considered in Sections 2, 3 and 4. In Section 2 we consider equivariant nonnali-zation from the point of view of normalinonnali-zation on orbit spaces. In Section 3 from the point of view of averaging over tori. In Section 4 we give a detailed treatment of further normalization of formal power series Hamiltonians with polynomial coefficients. In Section 5 it is shown how in the 3-dimensional case integrable normal forms can be reduced to one degree of freedom

systems. In Sections 6 and 7 the example of the lunar problem is considered.

A first example of further normalization is found in [2], where on an ad hoc basis a second order integrable approximation for the lunar problem was found. The general approach presented in this paper was developed when considering the orbiting dust model [5], which is another example to which the method of this paper applies.

§ 1. Preliminaries - a first normalization and reduction

In this section we review some known results about constrained normalization and reduc-tion of perturbed Keplerian systems. The main references are [1] and [4].

Let M c IR 2n be a submanifold given as the zero set of 2 m functions

11

= ... =

12m

= 0, m

<

n. Let ro be the standard symplectic form on 1R2/1, and suppose that M is a symplectic manifold with symplectic form roM = ro 1_{M, the restriction of ro to} M. The Hamil-tonian system on (1R2n, ro) with Hamiltonian function H : 1R2n ~ IR is denoted by (1R2n, ro, H),

and (IR 2n , ro, H) constrained to M is (M, roM, H 1_{M). Now let H} be a formal power series, that is, 00

H=

L

ek H/c..

/c.=O When

(c1) XHo has periodic flow

We may normalize H on 1R2n by using the algorithm of [1]. When in addition

(c2) M is invariant under the flow of XHo

H

1M

can be normalized on M, which we call constrained normalization (see[4]). The constrained normalization procedure comes down to the following. We start with normalizing H on 1R2n using the algorithm of [1]. At each step we adjust the normalizing transformation such that (1) it leaves M invariant and (2) it only changes the normal form by adding terms which vanish on M. As a consequence H

1M

is normalized on M by restricting the transformations to M.

The technique of constrained normalization can be applied to perturbed Keplerian systems of arbitrary dimeJlli· .1' ~.ve will restrict to the 3-dimensional case for convenience.

Consider IR I} ;.1.th coordinates (q, p) and standard symplectic form ro. Let

Ho(q,p)=(lp 121 q 12 _<q,p>2)II2,

Cs={(q,p)E IRS I Ho(q,p)=O}.

(1.1)

(1.2) On IR s \ C s consider a Hamiltonian system with formal power series Hamiltonian

H =

~

ek _{Hk , Ho as in (1.1) and Hk E Coo (IRs \C s), k}

~

1. Let;;' denote the restriction of ro to k=O

IRs \ Cs,let

(1.3) and let

~

be the restriction of ro to T+ S3. Then (T+ S3,~) is a symplectic manifold. Because

H 0 11" s] = I P I, the system (T+

S3,~,

H 0 Irs]) is precisely the regularized Kepler system for

negative energy (see [6]).

Proposition 1 [4] Each formal power series perturbation of a Keplerian system with negative energy can be written as a constrained system (T+ S3,~, H Irs]), where H is a formal power

series on IR s \ C s with H 0 as in (1.1), provided the perturbation is regularized together with the

Kepler system. (If not we have to exclude the collision set of the Kepler system from IR s \ C s

and T+ S3),

Proposition 2 [4] The flow of X_{Ho '} Ho as in (1.1), on IRs \ _{C s is periodic} and leaves T+ S3

invariant.

From Propositions 1 and 2 it is clear that we may apply constrained normalization to H

11"

S3

on T+ S3. The following proposition allows us to determine what such a normal form looks like

on IRs \ C s. Write F ::: G if F Irs3

=

G _Irs3, and let {{ , }} denote the Poisson bracket on

(T+ S3, ~).

Proposition 3 [1] {{ H 0 Irs3 , F Irs3 }}

=

0 if and only if F :::

p,

with

P

a formal power series of

A

which the coefficients FIc are smooth functions in the homogeneous quadratic polynomials

(1.4)

Corollary 4 Let

if

be a constrained normal form for H, up to order I, on IR s \ C s' then there exists

A _ A

an H ::: H such that HIc, 0$ k $ I, are smooth functions in the polynomials Sij' 1 $ i

<

j $ 4.

The S ij together with the Poisson bracket on IR S span a Lie algebra isomorphic to so( 4).

The action of the corresponding group SO(4) leaves H 0 . and T+ S3 invariant. This action

corresponds to the well known symmetry group of the Kepler system generated by the

momen-tum and Laplace vector.

Next we will consider reduction of Hamiltonian systems which on T+ S3 commute with

H 0, that is, which on IR S \ C S have a formal power series Hamiltonian with coefficients smooth

in the Sij.

Consider the map

By Propositions 2 and 3 the restriction of p to T+ S3 is an orbit map for the flow of XHo on

T+ S 3. Consequent! y M/ =

P

(T+ S 3 (') {H 0 = I}) are the reduced phase spaces for the X_H 0 -action

(cf. [1

D.

The image of p is determined by the relation

(1.6) Furthermore we have

H ( )2 ~ s?-. -/2

o

q,p

=

~ I) - • (1.7)

l:S;i <ft;4

The equations (1.6) and (1.7) completely determine the reduced phase space M/ as a 4-dimensional variety in 1R6. The coordinate change on 1R6 given by

XI = S 12

+

S 34 , X 2 = S 13 - S 24, X 3 = S 23

+

S 14 ,

Y 1 = S 12 - S 34, Y 2 = S 13

+

S 24, Y 3 = S 23 - S 14 ,

changes (1.6) and (1.7) to

(1.8)

(1.9) Consequently M_J is diffeomorphic to S2 x S2. Identifying JR.6 with so (4)* (* denoting dual) the linear coordinate change (1.8) is precisely the Lie algebra isomorphism between so(4)* and

(so(3)+so(3))*. The reduced phases spaces can be considered as co-adjoint orbits of SO (3) x SO (3) on the dual of its Lie algebra. The symplectic form on the reduced phase space corresponds to the Lie Poisson structure.

Now let H =

r.

£/cH/c, with H 0 as in (1.1) and Hb k

~

1, smooth in the Su. Then H = hop,

/c~

§ 2. Eguivariant nonnalization - normalization on the orbit space Consider instead of (1.5) the orbit map

p : T+ S3 ~ JR6; (q, p) H (X, y),

with X and Yas defined in (1.8). The XHo -orbit space p(T+S3) is given by the equation

Xt

+

xi

+

X~

=

Yt

+

Yi

+

Y~ , (X, Y) :;t

0 ,

(2.1)

(2.2)

which is equivalent to (1.6). Let A OO( JR 8) be the Lie subalgebra of COO(JR 8) which consists of the smooth functions in the quadratics Xj and Yj , i = 1 , 2, 3. For two functions F and G in A OO(JR 8)

we have

Next consider JR6 with coordinates (x, Y)

=

(Xl, x2, x3, Y l' Y2, Y3). Then P

=

P

(T+ S3) is

defined by P={(X,Y)E JR6Ixt+xi+x~=Yt+Yi+YL(x,y):;tO}. By a theorem of Schwarz [7] the pull back of p is surjective from

c

oo(JR6) onto A OO(JR 8). In fact for

f

E Coo(JR6)

let!

=

f

Ip , then by replacing (x, y) by (X, Y) !pulls back to

f

(X, Y) E A oo(JR8). It is now easy to

see that the Poisson structure on A OO(JR 8) given by the right hand side of (2.3) under p induces a Poisson structure on

c

oo(JR6) making p into a Poisson map. Because we have

{X_l ,X₂}=2X_{3 ,} {X_l ,X₃}=-2X_{2 ,} {X₂,X₃}=2X_{l ,}

{Y l ,Y2}=2Y3 , {Y l ,Y3}=-2Y2 , {Y2,Y3}=2Yl,{Xj,Yj}=O, we obtain on

c

oo(JR6) the Poisson bracket

which has a natural restriction to P.

(2.4)

(2.5)

Note that COO(P) is Poisson isomorphic with A OO(JR8)/I, where I is the ideal (under multi-plication) generated by the relation (2.2). In its tum A OO(JR8)/I can be identified with C""'(T+ S3)Ho, which is the space of Coo functions on T+ S3 invariant under the flow of XHo. Consequently eOO(T+ S3to and eOO(p) can be identified as Poisson algebras.

Given the Poisson structure (2.5) on JR6 we define for

f

E

c

oo(JR6) the Hamiltonian vector

Let h

=

ho

+

E h 1

+

E2 h2

+ ... ,

hi E Coo(JR6), be a formal power series. If Xho has periodic

flow then we may apply the normalization algorithm of [I] to normalize h.

Note that the fact that we are dealing with a Poisson structure instead of a symplectic struc-ture has no influence on the algorithm. Furthermore note that the Poisson bracket of any function

f

E JR 6 with xI

+

x~

+

x~ - YI - y~ - y~ vanishes. As a consequence the normal form on P is found by just restricting the normalization on JR6 to P.

Next consider on JR 8 a constrained normal form H corresponding to a perturbed Keplerian system, that is, H =H 0

+

E H 1

+

E2 H2

+ ... ,

with H 0 given by (1.2), and H" E A oo(JR8). We

can write H (q, p) = H(X, Y). Under pH corresponds to a Coo function on P which is precisely

H(x, y) Ip where H(x, y) is a smooth function on JR6 \ {OJ. Because Ho commutes (with respect to [-, -]) with every

f

E Coo(JR6 \ {O}) its flow exp XHo acts as the indentity. Consequently hex, y) = H(x, y) - H o(X, y) = E H 1 (x, y)

+

E2 H 2(X, y)

+ ...

and H(x, y) have equivalent flows

on JR6\ {OJ. Rescaling h gives hex, y)=H_{1 +EH 2(x, y)+ ..}. , which can now be normalized provided

Xii,

has periodic flow on 1R6 \ {O}.

Recall that we can move forth and back between A oo(JR 8) and

c

oo(JR6) by replacing (X, Y)

by (x, y). Similarly a symplectic transformation exp LF(q,p) , with LF = {-, F}, F E A oo(JR8) can be written as exp LF(X, Y) , F (q, p) = F(X, Y), which corresponds to a Poisson diffeomorphism exp Li(x, y) on JR 6. Consequently the normalization on the orbit space can be copied on JR 8. The normalizing transformations exp _{LF ,} FE A oo(JR8), are equivariant with respect to the flow of X_{Ho '} and the resulting normal form commutes with H 0 as well as with H 1.

On

c

oo(JR 6\{O}) with bracket [-,-] given by (2.5) we have that eachfE

c

oo(JR 6\{O})

commutes with H o (x, y) and XI

+

x~

+

X 3 2

- YI - y~ - y~. Consequently tJ,~ reduced phase

spaces MI are invariant under the normalizing transformation. Therefore the normalization on P

§ 3. Eguivariant nonnalization - averaging over Tori

Although not really necessary we restricted ourselves in the foregoing section to 3-dimensional perturbed Keplerian systems. In this section we will start from a more general point of view.

Let M be a symplectic submanifold of JR.2n as in Section 1, that is, M is given as the zero set of an even number of functions. By [4] a constrained nonnal fonn on M is the restriction to M of some fonnal power series

00

H = H 0 +

L

ek HIco Hk E ker LHo C COO

(JR.2n) .

k=1

(3.1)

Consider N°O(JR.2n) = {F E COO(JR.2n)1 {F, HoI =0, expLrF(M)eM , eE JR.I, that is,

N°O(JR.2n) is the space of Coo -functions in ker L_{Ho '} the flow of which leave M invariant. Lemma 5 N°O(JR.2n) is a Lie sub algebra of ker LHo.

Proof. Follows by using the Jacobi identity and Lemma 1 of [4].

Theorem 6. Let G : M ~ JR. be a Hamiltonian on M and let H be as in (3.1) such that H 1M is a constrained normal form for G. Then (exp Lr!F H) 1_{M ,} with F E N°O(JR. 2n ), is also a constrained nonnal fonn for G (with respect to H 0).

Thus we can use transfonnations exp Lu, F E Noo(JR.2n), to further normalize an obtained nor-mal form. We have

with and exp LrF H

=

H 0

+

~

ek Hk , k=!

-HI = {F, _{Ho} +HI =H I} H2=t {F, {F,Holl+{F,HJl+H2={F,Hd+H2 (3.2)

Let HIE N°O(JR.2n). We may restrict LHI to ker LHo C COO

(JR2n). If in addition there is a split-ting ker LHo = ker LHI ~ im LHI then it is clear from (3.2) that we may normalize H2 (con-strained) with respect to HI. Raising the power of e by one this process can be repeated to nor-malize H 3 etc.

The following theorem shows that under certain conditions a splitting

ker LHo = ker LHI ~ im LHI exists. It is a generalization of Proposition 1.1 in [1]. Recall that the flow of XH 0 is supposed to be periodic.

Theorem 7. Suppose H 0 and HI to be functionally independent If Mis fibered with 2-tori which are invariant under the flow of XHo and _{X H1 ,} and on which the flow of _{XH 1} is periodic or quasi-periodic, then there exists a splitting ker LHo = ker LHI ~ im _LH1•

Proof. Clearly there is a linear combination a H 0

+

~ HI, ~

*

0, which has periodic flow. Choose

a and ~

*

0 such that the period is minimal. The result of the theorem is now obtained by averag-ing over the periodic solutions of a H 0

+

~ HI as in [1]. It is obvious that this can be done

com-pletel y in ker LH o·

0

Note that H is supposed to be in constrained normal form with respect to H 0, that is H is obtained by averaging over the periodic orbits of X_H D. Of course both averaging processes can be

combined to one process of averaging over the invariant 2-tori.

In the case of 3-dimensional perturbed Keplerian systems N°O(JR2n) and leer LHo are replaced by ADO

(JR8). The conditions on the flow of X_H1 come down to the condition that the reduced XHI-flow must be periodic. As a consequence normalization on the orbit space P is, on

§ 4. Equivariant normalization - the case of polynomial coefficients Consider a fonnal power series

00

H=Ho+

L

fie Hie (4.1)

Ie=!

with Ho as in (1.1), and Hie E Aoo(JR8) polynomial. Let V_2n cAoo(JR8) be the set of polynomials

of degree 2n in (q, p) (and thus of degree n in (X, Y». Recall that V₂ is isomorphic to so(4).

(1 : F ~ LF, with LF

= {. ,

F}. is, upon a minus sign, the adjoint representation. Changing the

representation space from V 2 to V 2n we obtain just another algebraic representation of the adjoint representation. Consequently LF is semisimple on V 2n if and only if it is semisimple on V 2. If LF

acts as a semisimple linear operator on V 2n then we have V 2n = ker LF ED im LF. Thus we can

normalize H with respect to H I if we can show that LHI acts semisimply on V 2. The following

statement is obvious.

Theorem 7 LHI acts as a semisimple linear operator on V 2 if and only if

H! (q, p) =/(q, p) F(q, p) with 1 in the center of A oo(JR8) and FE V_{2 .}

Corollary 8 If H I is of the form given in Theorem 7 then H can be equivariantly normalized with

respect to HI'

Suppose H! is as in Theorem 7. We can write the factor F as

F

=

L

aj Xj

+

"f} Yj .

j=1

(4.2)

With REA OO(JR 8) we have exp LR HI

=

1 exp LR F. If REV 2 we may consider exp LR as being an element of SO (3) x SO (3) if we identify V 2 with so (3)

+

so (3). The action of exp LR is then identified with the co-adjoint action. Consequently we may put H! by a linear coordinate change in the form

-

-HI =1 (aX_I +yY!)=I· F

We have Liil = 1 Lj;. If we choose {X 1 , X 2, X 3 , Y 1 , Y 2, Y 3} as a basis for V 2 then the matrix of Lj; is 0 000

o

0 0 00.0

o

0

o

-a 0 0

o

0 0 000 00 (4.3) 0 000

o y

0

o

0 0

-y

0

which is semisimple. The matrix (4.3) is precisely the matrix of the vector field corresponding to

F

on the orbit space P = p(T+ S3), with respect to the Poisson structure (2.5). The corresponding flow on the orbit space is periodic if 0./ Y E {2 (which corresponds to the cases considered in

Sections 2 and 3), and densely fills a 2-torus if o./y is irrational. We will call these the resonant and the nonresonant case respectively. We may consider the reduced system corresponding to HI

as

two coupled harmonic oscillators.

We can thus determine ker Lj; in a straightforward way. In the nonresonant case ker Lj; is generated by

(4.4) In the resonant case we may without loss of generality restrict ourselves to the case

0. E IN \ {O} , Y E Z \ {O}, g.c.d. (0., y)

=

1. In order to compute the kernel of Lj; we introduce complex conjugate variables Zl =X2 +iX3' Z2 = Y2

+

i Y3, '1 =11 =X2 -iX3, and '2 =12 = Y2 - i Y3 • We obtain

Lj; =-i o.(Zl

~-'1 ~)-

iJ<Z2

~-'2~)

ih

1 a'l aZ 2 a'2 It is now easily found that ker Lj; is generated by

1t1 =X 1, 1t2 = Y 1, 1t3 = Z1 '1 =xy + X~ , 1t4 = Z2'2 = Y~ + Y~ ,

and

(4.6) gives rise to the real generators

.

{ t

(zI

'~+'I z~),

1ts = ..!. (zlyl ZU +ylyl YU)

2 1 2 '01 '02 ,

ify> 0, ify< O.

In both cases we have among the generators the relation

.

.

(4.5) (4.6) (4.7) (4.8) (4.9)

Provided H I fulfills the conditions of Theorem 7 this characterizes the terms that will appear in the normal form.

Note that by the results of Schwarz [7] A OO(R 8) (") ker LHI consists of all smooth functions in the generators 1t1, ... ,1t(;. Consequently in the resonant case, that is, X_HI has periodic flow,

§ 5. Reduction to one degree of freedom Consider a Hamiltonian H

=

H 0

+

e H 1

+

L

ell H II (5.1) 11=2 with H 0 give by (1.1), HI

=

(m Xl ± k Y 1) I ,

where m, kEN \ {O}, g.c.d. (m, k) = 1 and I is in the center of A OO(m8), and with {HII' H o} = {HII , HI}

=

0, n ~ 2. Thus we suppose H to be in normal form with respect to H 0

as well as H 1. Consequently the coefficients _{HII ,}

n

~ 2,

are

smooth function in the generators 1tl, ... ,1t(j given in (4.5), (4.7) and (4.8).

In Section 1 we obtained a first reduced phase space by using the orbit map for the XHo -flow on T+ S3 , which is given by

p : T+ S3 ~ 1R6 ; (q, p) H (X, Y). (5.2)

The orbit space P

=

p(T+ S3) is defined by the equation (2.2), i.e. XI

+

X~

+

X~

=

YI

+

Y~

+

Y~. The reduced phase spaces MI

=

p(T+ S3 II (Ho=l})

are

given by equations (1.9), i.e. XI

+

X~

+

X~

=

[2, and Yy

+

Y~

+

Y~ = [2. Writing H =

if

0 p we obtain in a trivial way the

reduced Hamiltonian H on MI. On MI we have the symplectic structure induced by the Poisson structure (2.5).

The flow of the reduced vector field Xii, on M is periodic and [H, H

d

=

O. Thus we may apply reduction with respect to the Xii, flow.

The orbit n .p ~or the Xii, -flow on MI is given by

(5.3) The orbit space PI = _(MI) is determined by the following equations and inequalities obtained from (1.9), (4,5), and (4.9)

7tI

+

1t3

=

[2 , 1t~

+

1t4

=

[2 ,

7t~

+

1t~ = 1t~

1tT,

1t3 ~ 0, 1t4 ~ 0 . (5.4)

The reduced phase spaces PI,c

are

given by (5.4) and

m 7t1

±

k 7t2 = C • (5.5)

Because I _. is constant on MI (5.5) is equivalent to HI = constant The reduced phase spaces PI _, c are 2-dimensional semi-algebraic varieties. From (5.4) and (5.5) we obtain the following descrip-tion of PI,c in (7ts. 1t(j. 7tl)-space

(5.6)

-kl+c kl+c

-I ~ 1tl ~ 1 , ~ 1tl ~ -'--'--- (5.7)

m

m

which holds for the plus as well as for the minus sign in (5.5).

Equation (5.6) describes a surface of revolution. The inequalities (5.7) restrict to a part of this surface. The bounds for 1tl are zeroes of the right hand side of (5.6). As a consequence the different types of reduces phase spaces P1,c are distinguished by the position of the zeroes of the right hand side of (5.6) relative to each other, and their multiplicities. All the possibilities are listed in Tables I, II, III, together with the corresponding bounds for the parameter c. We have to distinguish between the cases 0

<

m

<

k (Table I), 0

<

k

<

m (Table II), and m

=

k

=

1 (Table

III).

We obtain that for - (k

+

m) 1

<

c

<

(k

+

m) 1 the reduced phase spaces are obtained by rotating around the 1tl-axis the part of the graph of the right hand side of (5.6) which lies in between the two middle roots.

The reduced phase spaces tum out to be sphere like surfaces which we denote by S (s, n)

here s is the multiplicity of the smallest root (south pole), and n is the multiplicity of the largest root (north pole). In fact S (s, n) is a topological 2-sphere which is smooth except for two cusp like singularities, one of contact order (n - 2) at the north pole and one of contact order (s - 2) at the south pole. SO, 1) is a smooth two sphere, while n

=

2 or s

=

2 gives a cone-like singularity.

An important fact to notice is that the nature of the reduced phase space differs with the parame-ter c (= energy of Xii.), and that in general the reduced phase space is not a differentiable mani-fold.

TABLE I 0

<

m

<

k

bounds fore ordering of zeroes respective multiplicities reduced phase space -kl+e kl+e 1 1 e < -(m

+

k)1 < - - < - < m,m, k,k

o

m m e

=

-(m

+

k)1 - k 1

+

e

<

k 1

+

C

=

-I

<

1 m m m,m+k,k point -kl+c I kl+c 1 -(m+k)/<c«m-k)1 <- < - - < m,k,m,k S(k, m) m m c

=

(m - k)1 - k 1

+

e

<

-I k 1

+

C

=

1 _{m,k,m+k S(k, m +k)}

m

m

-kl+c 1 1 kl+c (m-k)1 <c < (k-m)1 <- < - - m,k, k,m S (k, k)

m

m

c = (k - m)l - k 1

+

C =

<

-I

<

1 k 1

+

C _{m+k,k,m S(m+k, k)} m m (k - m)1 < c < (k

+

m)1 -I < - k 1

+

c < 1 < 1 k /

+

c k,m, k,m S(m, k) m m e

=

(k

+

m)1 -I

< -

k 1

+

e

=

I

<

k 1

+

c m m k,m+k,m point 1 1 -kl+c kl+c e

>

(k +m)/ - < < < - - k, k,m, m

o

m m T ABLE II 0

<

k

<

m

boundsforc ordering of zeroes respective multiplicities reduced phase space c < -(m +k)1 -kl+c < - -kl+c <-I <I m, m, k,k 0

m m

c =-(m+k)1 -kl+c < --=-/<1 m,m+k,k point

m 1

-(m+k)1 < (k-m)1 -kl+c <-I < - - <I . kl+e m,k,m,k S(k, m)

m m c

=

(k-m)1 -kl+c =-1 < kl+c <I m+k,m,k S(m+k, m) m m (k-m)1 < c < (m -k)1 -1< -kl+c < - - < k I+c 1 k,m,m,k S(m, m) m m c = (m-k)1 -1< -kl+c < kl+e =1 k,m,m+k S(m, m +k) m m (m -k)l < e < (m +k)1 -1< -kl+c < < - -I kl+c k,m,k,m S(m, k) m m c = (m+k)1 -1< -kl+c =1 < kl+c k,m+k,m point m m c

>

(m+k)l -1<1< -kl+c < - -kl+c k,k,m,m 0 m m

TABLEIII k=m=1

boundsforc ordering of zeroes respective multiplicities reduced phase space

c <-21 -1+c<l+c<-I<1 1 , 1, 1 , 1 0

c =-21 -I+c <I+c=-I<I 1,2, 1 point

-2/<c<O -1+c<-I<I+c<1 1,1,1,1 S(1,1)

c=O -1=-I<L=1 2,2 S(2,2)

O<c<2L -L <-L+c < 1 < L+c 1,2, 1 S (1, 1)

c =2 L -1 <-I+c =L < l+c 1,2, 1 point

§ 6. The lunar problem - normal form

In this section we will show that the theory of the previous sections applies to the three dimensional lunar problem. We obtain a second order integrable normal form for the lunar prob-lem which can be analyzed in a straightforward way. Our results partly cover earlier results of Kummer [2] who also analyzed a second order integrable normal form for the lunar problem. We will not compare the two normal forms because in general normal forms are not unique ([3]) and it can be quite hard to find the diffeomorhpism mapping two given normal forms to each other.

The Hamiltonian for the lunar problem is given by K(x,y)=.!. Iy

12-_1_-A.(XIY2-X2Yl)-2 1 xl

_1..2 (I-v) (3xr _I x 12)

+

o

(v-l 1..4)

2 (6.1)

which we consider on the energy surface K

=

-.!. k2' k

>

O. A. is the perturbation parameter and

2

v is the relative earth mass. After regularization and constrained normalization we obtain (see [4] corrected version), up to order two, in terms of Xi , Y_{i ,} i

=

1 , 2, 3,

H (q, p)

=

H o(q, p)

+

A. H 1 (q, p)

+

1..2 H 2(q, p) with H o(q. p) given by (l.1), and

1 HI

= -

2k H 0 (X 1

+

Y 1 ) H2

=-~

(I

~v)

_{Ho (X 3} -y_{3)2 +.!.} (I-v) H3 + 1 H (X +Y

i

4 k 2 k3 0 16 k3 0 1 1 3 (I-v) 2

+

16

k3 Ho

(2Ho-2(X3Y3+X2Y2+XIY1»)-_.l....

(l

~v)

Ho (2H5 +2(X lY 1 _{+X2Y2 -X3Y} 3» 16 k (6.2) 1 2 (.!. + _1_) (X 1 +Y r)2 _{(2H5 -2(X3Y3 +X2Y} 2 +X lY 1» (6.3) 128 k 2 Ho where 3 2 H5 =

L

Xt

+

Yt . i=1

For further normalization of H 2 we may apply the theory of Section 4. From (6.2) we see that we have a 1 : 1 resonance. Thus ker LH I = ker Lx I + Y I is generated by

1t}

=

X } , 1t2

=

Y 1 , 1t3

=

X~

+

X~ , 1t4

=

Y~

+

Y~ , 1ts =X 2Y 2 +X3Y 3 , 1t6 =X2Y3 -X3Y 2 ,

with the relation

(6.5)

To obtain the second order nonnal fonn with respect to H 0 and H I we have to split H 2 in a part in ker LHI and a part in _{im L H1 ,} the part in kerLHI then is the desired nonnal form. We have

LX1+Y1 (X 3Y 2)=- 2X2 Y2 + 2X 3 Y3,

L_X1+_Y1 (X2X3)=2X~ -2X~ ,

L_X1+_Y2 (Y2Y3)=2 Y~ -2 Y~

Consequently we obtain the following splitting for X~ , Y~ , X 3 Y 3,

X~ =.!.. (X~ +X~) -.!.. (X~ -X~)

2 2

Y~

=t

(Y~ +Y~)

-t

(Y~ -Y~),

X3 Y3

=t

(X 2Y 2+X 3y3

)-t

(X 2Y 2 -X 3Y 3).

Thus the normal fonn for H 2 is

3 (I-v) 2 2 3 (I-v)

- 8"

k₃ Ho (Y2 +Y3)+

8"

-k-_{3 -} Ho (X 2Y 2+ X 3Y 3)

-1. (I-;V) HoXIYI-_1_2 (.l.+_1_)H5(x_l+y_l)2

4 k 64k 2 Ho

+ _1_2 (.!.. +_1_) (XI +y₁)2 _{(X2Y2 +X3Y3)+}

64 k 2 Ho

+_1-2 (.!..+_1-)(X₁+y_{l)2 XI}

Y~

64 k 2 Ho

The reduced phase spaces PI.c (see Section 5) are given by

H 0 = [ , X I + Y I = 2c ,

lti

+ 1t3 = [2 , x~ + X4 = [2 • x~ + xg = X3X4.

We get that on P1•c our Hamiltonian is, modulo constants, equal to

- 2 H 2

=

a (XI - c)

+

~ 7ts • with a =

1.

(I-v) 1-_1_ (.!..

+.1)

c2 • 2 k3 16 k2 2 I

13=1.

(I-v) 1+_I_(.!..+.l)c2 . 8 k3 _{16 k}2 _{2 I}

In fact (6.7) gives the reduced Hamiltonian on P_1•c parametrized in (xs .1t6, xI)-space.

(6.6)

(6.7)

§ 7. The lunar problem - analysis of the integrable approximation

In the previous section we obtained a normal form for the lunar problem up to order two. Truncation gave us an integrable approximation of the lunar problem. Applying reduction as in Section S we obtain in (1ts, 1t6, 1tl)-space the reduced phase space PI,e given by (see (S.6) and (S.7».

-I ~ 1tl ~ I, 2c -I ~ 1t~ 1+ 2c, I c I ~ I, I > 0 .

and the reduced Hamiltonian (see (6.7) and (6.8»

Substituting crl = 1tl - C , cr2 = 1ts , cr3 = 1t6 we get I crl I ~ I - I c I , I c I ~ 1 , l

>

0 ,

H

2

=

a. cry

+

~ cr2 . (7.1) (7.2) (7.3) (7.4) (7.S) (7.6) From Table III we see that the reduced phase space PI,e is S(l, 1), that is, a smooth S2, for

o

<

I c I

<

I, and S (2, 2) for c

=

O. The reduced system is a one degree of freedom system. The trajectories are precisely the intersections of the H 2 level surfaces with the two dimensional reduced phase space PI,e' We know the global phase portrait if we know the critical points of H 2 on PI,e' These critical points correspond to the stationary points of the reduced system. These crit-ical points were determined for general a. and ~ in [S]. We will not repeat the analysis but state the results using [S].We have to take care of the fact that in the case of the lunar problem a. and ~

depend on c, that is, the sign of 0.2 - ~2 might change with c. Furthermore ~

*"

0, but a can be equal to zero.

H 2 has a critical point on PI,e if the energy surface H 2

=

L is tangent to PI,e' Using Lagrange multipliers it is easily obtained that all critical points must be in the cr3

=

0 plane, that is, on the topological circle sl,e = Pl,e r'I {cr3 = O}. Putting cr3 = 0 in (7.4) and eliminating cr2 Using h

=

a. crt

+

~ cr2 , ~

*"

0, we obtain

(0.2 - ~2) cr1

+

2(-o.h

+

~2(l2+c2» crt

+

(h2 - ~2(l2-c2)2)

=

0

I crl I ~ 1 - I

c

I, I

c

I ~ I , 1

>

0 .

(7.7) (7.8) Now (crl,

~

(h -a. cry), 0) is critical point of

H2

on PI,e if and only if crl is a double root of (7.7) which satisfies (7.8). Considering the discriminant locus of (7.7) taking the inequalities (7.8) into account gives that (7.7) has three branches of double roots given by (see [S])

h =±(l (l-v)

I+_l_(~

+.l)C2 ) (l2-C 2), I C lSI, I > 0

8 k3 _16k2 _{2 I} (7.9)

h=(l (I-v) I _ _ l_(~ +.l)C2 ) (l2+c 2)_

2 k3 16k2 2 I

-211 C 1 _/135 (I-v)

I2_~

(I-v) _{(i+ 1)c2 ,}

'I

64 k6 _{64 k}5 ₂

I ciS I

~

I8(I-v) , I> 0

30(l-v)+k(l +2) (7.10)

Note that for c

=

Co

=

I

~

18(1-v) ; c

=

-Co the third branch attaches to the positive

30(I-v) + k(I +2)

branch of (7.9). The third branch only exists for 0.2 - ~2 ~ 0, which is equivalent to

I c I

<

I

~ ~8(~1:2v/

. Because Co

<

I

~ ~8g:;/

this last condition is always satisfied. The tangency of h

=

0.

crt

+ ~ Cf2 and S

f.e

(in the (Cfl, Cf2)-plane) is sketched in Figure 1 for

the different values of c. In Figure 2 the families of critical point are given in the parameter plane

(c, h).

From Figure 1 it is immediately clear which critical points are elliptic and which are hyper-bolic. This is indicated in Figure 2. Note that for the branch attached to ho (corresponds to (7.10»

each point corresponds to two critical points (see Fig. lb.).

The results found are an extension of and in agreement with the results in [2]. Note that the points (± I, 0) and (0, h

_o)

(counted twice) corresponds to the four critical points of the only once reduced system on MI'

For a further discussion of how the results found here are related to the luru' . :~..,.,blem in its original fonnulation we refer the reader to [2].

a: c

o

b: c: 1 d:

I

1=i!.(480-V 2

]2

c k(i!.+22 1 e:

I

1>i!.[48Cl-V2J2 c k(i!.+22

figure 1. Tangency of

H2

and Sl _{i!. , c}

h 1.

C _ (

18C1-v)

]2

o-i!. 30C1-v2+kCi!.+22

c

References

1. R. Cushman (1984), Normal forms for Hamiltonian vector fields with periodic flow. In: Dif-ferential geometric methods in Mathematical physics, p. 125 - 144, ed. S. Sternberg, Reidel, Dordrecht, 1984.

2. M. Kummer (1983), On the 3-dimensionallunar problem and other perturbation problems of the Kepler problem. J. Math. An. Appl. 93, 142 - 194.

3. J.C. van der Meer (1985), The Hamiltonian Hopf bifurcation, Lecture Notes in Mathematics 1160, Springer, Berlin etc., 1985.

4. J.C. van der Meer, R. Cushman (1986), Constrained normalization of Hamiltonian systems and perturbed Keplerian motion. Z. Angew. Math. Phys. 37, 402 - 424; Corrections ibid 37,931.

5. J.e. van der Meer, R. Cushman (1986), Orbiting dust under radiation pressure, In: Proceed-ings of the XV'h Inl Conf. on Diff. Geometric Meth. in Theoretical Phys, Oausthal, 1986, 403 - 414, World Scientific, Singapore, 1987.

6. J. Moser (1970), Regularization of Kepler's problem and the averaging method on a mani-fold, Comm. Pure Appl. Math. 23,616 - 636.

7. G. Schwarz (1975), Smooth junctions invariant under the action of a compact Lie group,

Topology 14, 63 - 68.

8. A. Weinstein (1983), The local structure of Poisson manifolds, J. Diff. Geometry 18, 523 -557.