Partial reduction in the N-body planetary problem using the

Nov 15, 2001 - manipulate the perturbative series. 3. Reduction in the three-body problem. In the particular case of three bodies, Jacobi's reduction can be per-.
842KB taille 5 téléchargements 444 vues
Partial reduction in the N-body planetary problem using the angular momentum integral F. Malige ([email protected]), P. Robutel ([email protected]) and J. Laskar ([email protected]) Astronomie et Syst`emes Dynamiques, IMCCE, CNRS UMR8028, 77 Av. Denfert-Rochereau, 75014 Paris, France. November 15, 2001 Abstract. We present a new set of variables for the reduction of the planetary n-body problem, associated to the angular momentum integral, which can be of any use for perturbation theory. The construction of these variables is performed in two steps. A first reduction, called partial is based only on the fixed direction of the angular momentum. The reduction can then be completed using the norm of the angular momentum. In fact, the partial reduction presents many advantages. In particular, We keep some symmetries in the equations of motion (d’Alembert relations). Moreover, in the reduced secular system, we can construct a Birkhoff normal form at any order. Finally, the topology of this problem remains the same as for the non-reduced system, contrarily to the Jacobi’s reduction where a singularity is present for zero inclinations. For three bodies, these reductions can be done in a very simple way in Poincar´e’s rectangular variables. In the general n-body case, the reduction can be performed up to a fixed degree in eccentricities and inclinations, using computer algebra expansions. As an example, we provide the truncated expressions for the change of variable in the 4-body case, obtained using the computer algebra system TRIP. Keywords: Perturbation Theory, Reduction, Jacobi’s Reduction, Resonance, Lie Transformation, Computer Algebra

1. Introduction The n-body problem possesses seven well-known integrals (energy, linear and angular momentum) and many ways have been found to reduce the number of equations of motion (6n equations for n bodies) using these integrals since (Lagrange, 1772). The theoretical reduction is presented in (Meyer and Hall, 1992) and (Arnold et al., 1985). For the n-body planetary problem, the difficulty remains to find practically the simplest set of variables of the reduced problem which could be effectively used for perturbation methods. To achieve the linear momentum reduction (section 2), Poincar´e proposed two systems of variables, both defined by a linear map from the Euclidean coordinates: the canonical heliocentric variables and Jacobi’s coordinates (Poincar´e, 1896 see also Laskar, 1989a). These two reductions present various advantages, and c 2001 Kluwer Academic Publishers. Printed in the Netherlands. °

articleMRL2.tex; 15/11/2001; 17:51; p.1

2 the choice of one or the other will depend on the specific problem that is studied. In both cases, only 6n − 6 variables remain in the reduced system. Regarding the angular momentum, the situation is more complicated, as no practical reduction for the general case of n bodies by a simple change of variables is known, except in the case of three bodies where there exists a simple reduction, called ”elimination of the nodes” (Jacobi, 1842). Bennett proposed a method to perform the reduction for n bodies, but this method is very difficult (Bennett, 1905), (Hagihara, 1970). Since, a succession of works (Boigey, 1981), (Deprit, 1983), using a chain of kinetic frames proposed an explicit reduction for 4 bodies and then for n bodies which is simpler than Bennett’s one but which is still difficult to implement. Moreover, in these studies, the reduced variables do not have a simple physical meaning and to express the Hamiltonian of the system by means of these variables is quite difficult. In the present work, we propose a new method based on a ”partial” reduction of the problem. Let C be the total angular momentum of the system. If we use the invariant plane (orthogonal to C) as the reference plane of the system, the two components of C in this plane, Cx and Cy , are equal to zero. The partial reduction is a reduction where only the direction of C is taken into account, and not its norm. In the case of three bodies, in the invariant plane reference frame, the angles of the ascending nodes are opposed, and the partial reduction can be done easily (section 3). In the general n-body case, we will use the integrals Cx = Cy = 0 to express two variables as functions of the other ones. This transformation is not canonical and we loose the standard form of the Hamilton’s equations. Nevertheless, we can then make an additional (non-canonical) transformation in order to recover the standard form of the symplectic form, using a constructive version of Darboux’s theorem (section 4). This step is computer assisted and can only be done up to a given degree in eccentricity and inclination. There are several motivations to search for such a transformation. The first one is to obtain a system with less degrees of freedom than the original one and which can thus be more easily studied. Secondly, the reduction enables us to prove that the construction of a Birkhoff normal form for the secular system can be formally done at any order and that no resonances will appear in this construction (section 5). A more topological reason is that the fixed point of the secular problem at the origin (planar and circular trajectories) remains the same in the partially reduced problem. The problem does not present the singularity encountered in the Jacobi’s reduction when the inclinations tend to zero (Robutel, 1995). We finally remark that once the partial

articleMRL2.tex; 15/11/2001; 17:51; p.2

3 reduction is done, a complete reduction, using the norm of the angular momentum C can easily be achieved (section 4).

2. Reduction by the invariance of the linear momentum This part is largely inspired by (Laskar, 1989a) or (Laskar and Robutel, 1995) which can be consulted for more details. Let P0 , P1 , . . . , Pn be n + 1 bodies of masses m0 , m1 , . . . , mn in Newtonian gravitational interaction, and let O be their barycenter. We assume that we are in the case of a planetary system where P0 is a massive body (for all j, 1 ≤ j ≤ n, m0 À mj ) and the other bodies ”turn” around P0 with quasi circular trajectories, nearly in the same ~ j . In the barycentric reference plane. For every body Pj , let uj = OP frame with origin O, Newton’s equations of motion form a differential system of order 6(n + 1) and can be written in Hamiltonian form using the canonical coordinates (uj , u ˜ j = mj u˙ j )j=0,n with Hamiltonian n

u j k2 1 X k˜ −G H= 2 j=0 mj

X

0≤j 2) In the general case of n planets (n > 2), the previous reduction cannot be done. Indeed, the opposition of the nodes (Ω1 = Ω2 + π) is essential in this reduction. In the general case, there is no such linear relation between the nodes, and an easy generalization of the precedent method is not possible. However, the partial reduction can still be achieved through a different method. 4.1.

Partial reduction and standard symplectic form.

The integral C1 (8) allows to express yn in term of the other variables. In the second degree equation of unknown |yn |2 obtained from C1 = 0,

articleMRL2.tex; 15/11/2001; 17:51; p.12

13 we retain for yn the root which tends to zero with the (yj )(1≤j≤n−1) √ 2α yn = r (35) q 2 2 2 2 Λn − |xn | + (Λn − |xn | ) − 2|α| with

α=−

n−1 X

yj

j=1

s

|yj |2 Λj − |xj | − . 2 2

(36)

We then replace yn (and y¯n ) by its expression (35) in the Hamiltonian. As (35) satisfies d’Alembert relations, it will be the same for the Hamiltonian. Practically, all quantities are expanded in series up to a fixed degree d in the variables xj , yj . Unfortunately, the restriction of the standard symplectic form (6) on the submanifold C1 = 0 is no longer standard so the Hamiltonian standard form of the equations of motion is not conserved. We have now σ=

n X k=1

[dλk ∧ dΛk − i(dxk ∧ d¯ xk )] −

n−1 X k=1

i(dyk ∧ d¯ yk ) − idfy ∧ df¯y (37)

where yn = fy (x, y, Λ) is expanded in series in the variables x, y (with coefficients depending on Λ)2 , the symplectic form σ can thus be written as an infinite serie: +∞ X σ= σk (38) k=0

where σk is the part of the 2-form σ which coefficients are homegeneous polynomials of degree k in the variables x, y. At lower order in the variables x, y, we have (35): s n−1 X Λi . (39) yn = − yi Λ n i=1

The first term σ0 of the 2-form consists of the terms which coefficients do not depend on x, y, that is σ0 =

n X k=1

[dλk ∧ dΛk − i(dxk ∧ d¯ xk )] − −i

n−1 X

j,k=1

n−1 X k=1

p Λj Λk dyj ∧ d¯ yk Λn

i(dyk ∧ d¯ yk ) (40)

2

In the following, x = (x1 , . . . , xn ), y = (y1 , . . . , yn−1 ), Λ = (Λ1 , . . . , Λn ), λ = (λ1 , . . . , λn ).

articleMRL2.tex; 15/11/2001; 17:51; p.13

14 It is thus clear that even at lower order in x, y, the symplectic form is not standard. We will now construct a non-symplectic transformation, degree by degree in x, y, in order to recover the standard form of this symplectic 2-form. At order 0, this will be achieved through a linear transformation. At higher order, we will use a constructive version of Darboux’s theorem to set up a near identity transformation which bring the symplectic form back to a standard form at all degrees in x, y. Remark. In the symplectic form σ, the terms dλ are only present in σ0 as fy does not depend of the λj .

4.1.1. Reduction at order 0 In the coordinate system (λ, x, y, Λ, −i¯ x, −i¯ y ), the matrix of σ0 is   0 0 I2n 0  0 0 0 A   (41)  −I2n 0 0 0  0 −A 0 0

where I2n is the (2n, 2n) identity matrix and A = In−1 + Vt V is a (n − 1, n − 1) real symmetric matrix with   √ √ Λ1 /√Λn √  Λ2 / Λn    (42) V=  . ..  . √  √ Λn−1 / Λn

The symplectic form σ0 is already standard in x, λ, Λ. We will thus make a simple linear transformation y = PY

(43)

such that t P AP = In−1 , where P is a (n−1, n−1) real matrix, and Y = (Y1 , . . . , Yn−1 ) are the new variables. In fact, the particular structure of the matrix A allows to find this transformation in an explicit way. Let (ej )j=1,n−1 be a canonical basis of Rn−1 , and Q an orthogonal matrix which brings en−1 (for example) on the normalized vector V/ kVk. In the new basis (Q ej )j=1,n−1 , the matrix of the quadratic form defined by A is diagonal (that is diag(1, 1, . . . , 1, 1 + kVk2 )). The q

affinity D of matrix diag(1, 1, . . . , 1, 1/ 1 + kVk2 ) will bring the matrix of the quadratic form to the identity. As the set of the possible matrix Q is isomorphic to On−2 , we have many choices. A simple choice

articleMRL2.tex; 15/11/2001; 17:51; p.14

15 is given by the orthogonal symmetry with respect to the hyperplane orthogonal to v = V/ kVk − en−1 , which gives, for u ∈ Rn−1 Q u = u − 2 < u, v >

v kvk2

(44)

where < u, v > is the usual scalar product in Rn−1 . The explicit expression of Q is given in the appendix, as well as an alternate transformation, obtained by a Schmidt’s orthogonalisation. We define P1 the transformation     x x −1   y  →Y =P y (45)   Λ Λ λ λ

It should be noted that this transformation is linear and thus conserves the d’Alembert relations in the Hamiltonian. Remark. Although formally it will not make any difference, in practice, it could be useful to choose yn such that the matrix A (Eq. 41, 42) is close to identity. The best choice in this case is to choose for yn the planet with the largest value of Λ, that is the variable associated to the body which contributes the most to the normal component of the angular momentum Cz . In some sense, this is similar to what is done for the linear momentum reduction when the reference frame is centered on the more massive body (the Sun).

4.1.2. Higher orders 4.1.2.1. Theoretical part. All the notations here are taken from (Spivak, 1999). We are looking for new variables p˜ in which the symplectic form will be standard. In order to calculate the transformation between p˜ and the old variables p, we use a constructive version of Darboux’s theorem. We construct a differentiable path in the set of the symplectic forms represented by the map t 7→ σt , t ∈ [0, 1] such that σ0 is standard and σ1 = σ is our symplectic form. Thus, We are looking for a transformation φt that satisfies, ∀t ∈ [0, 1], [σt ]φt (˜p) ((φt ∗ )p˜(v1 ), (φt ∗ )p˜(v2 )) = [σ0 ]φ0 (˜p) ((φ0 ∗ )p˜(v1 ), (φ0 ∗ )p˜(v2 )) (46) where (v1 , v2 ) are any couple of vectors in Tp˜R6n−2 , where (φ∗ )p˜(v) = Dφ(˜ p)(v)

(47)

articleMRL2.tex; 15/11/2001; 17:51; p.15

16 is the tangent map at the point p˜ associated to the diffeomorphism φ (v is a vector of Tp˜R6n−2 ) and such that φ0 (˜ p) = p˜ and φ1 (˜ p) = p. So, for t = 1, we will have [σ]p ((φ1 ∗ )p˜(v1 ), (φ1 ∗ )p˜(v2 )) = [σ0 ]p˜(v1 , v2 )

(48)

And our symplectic form will be standard in the variables p˜. We thus are looking for φt which pull back the symplectic form σt at the standard form σ0 for all values of t ∈ [0, 1]. The pull back of a two-form α by a diffeomorphism ψ is defined by [ψ ∗ α]p˜(v1 , v2 ) = [α]ψ(˜p) ((ψ∗ )p˜(v1 ), (ψ∗ )p˜(v2 )).

(49)

Thus we want that [φ∗t σt ]p˜(v1 , v2 ) = [φ∗0 σ0 ]p˜(v1 , v2 )

(50)

In order to find the vector field Xt associated to the flow φt , we derive (50) with respect to t: d ∗ φ σt = 0 . (51) dt t We have d ∗ φ∗ σt+s − φ∗t σt φt σt = lim ( t+s ) s→0 dt s = lim ( s→0

φ∗t+s σt+s − φ∗t σt+s φ∗t σt+s − φ∗t σt + ) s s = φ∗t LXt (σt ) + φ∗t

∂σt ∂t

d | φ∗ α is the Lie’s derivative following Xt of the ds s=0 t+s two-form α. So, as φ∗t is a linear map, where LXt (α) =

φ∗t (LXt (σt ) +

∂ (σt )) = 0 . ∂t

(52)

And thus, as φt is a diffeomorphism, LXt (σt ) +

∂ (σt ) = 0 . ∂t

(53)

So, by Cartan’s formulae, Xt dσt + d(Xt σt ) +

∂ (σt ) = 0 . ∂t

(54)

articleMRL2.tex; 15/11/2001; 17:51; p.16

17 where Xt dσt represents the inner product of Xt into dσt . Let t 7→ θt , t ∈ [0, 1] be a differentiable path such that for all t, θt is a real primitive of σt . This primitive has to be real to have a transformation such that the new variables associated to x, x ¯ are still complex conjugates. Owing to the fact that σt = dθt and dσt = 0, we have d(Xt σt +

∂ θt ) = 0 . ∂t

(55)

So, for any function g defined on the symplectic manifold, we have an equation satisfied by Xt : Xt σt = −

∂ θt + dg . ∂t

(56)

σt being non degenerated, once a function g is chosen, there exists a unique Xt such that this equation holds. Expressed in the coordinates (λ, x, Y, Λ, −i¯ x, −iY¯ )3 , we obtain in matricial notation Xt =

∂ (Mσt )−1 (− (t θt )) ∂t {z } |

transf ormation to make the 2−f orm standard

+

(M )−1 (t dg) } | σt {z

Hamiltonian vector f ield

(57) where Mσt is the matrix of the symplectic form σt in the chosen coordinates. We have thus unicity of Xt modulo an arbitrary Hamiltonian vector field. Practically, we take g = 0, as g 6= 0 corresponds to perform an additional symplectic transformation which does not change the symplectic form. We have made some attempts to find a more appropriate function g (which for instance leaves invariant the x variables as in the three-body problem partial reduction) but this did not led to any significant results. To define practically the differentiable path, we take the following real primitive of the symplectic form : n

1X xj + i¯ xj dxj ) θ1 = [ (λj dΛj − Λj dλj − ixj d¯ 2 j=1 −i

n−1 X j=1

(yj d¯ yj − y¯j dyj ) − i(fy df¯y − f¯y dfy )]

(58)

(we have dθ1 = σ1 ). When this one-form is expressed in the coordinates (x,Y ), it becomes an infinite series in these new variables. We then define the path on the primitive 3

In the following Y = (Y1 , . . . , Yn−1 ).

articleMRL2.tex; 15/11/2001; 17:51; p.17

18 θt = θ0 + t(θ1 − θ0 )

(59)

where n−1 n n X X 1X (Yj dY¯j − Y¯j dYj )] (xj d¯ xj − x ¯j dxj )−i θ0 = [ (λj dΛj −Λj dλj )−i 2 j=1 j=1 j=1

(60) We take σt = dθt . Thus, the map t 7→ σt , t ∈ [0, 1] satisfies the wanted properties.

4.1.2.2. Computational part. In order to compute the vector field Xt , we compute the inverse of the symplectic form up to a given degree d by the formal expansion X −1 k Mσ−1 = (Id − J −1 [J − Mσt ]) J −1 = [J −1 (J − Mσt )] J −1 (61) t k≥0

where J is the matrix of the standard symplectic form σ0 µ ¶ 0 I3n−1 J= . −I3n−1 0

(62)

As Mσt − J is at least of degree one in the variables x, Y , the sum (61) is only computed up to k ≤ d. The vector field Xt is then computed through (57), and the transformation φt is obtained with a Lie method. Let x be the old variables and x ˜ the new ones. We search a function φt (˜ x) such that φ0 (˜ x) = x ˜ and φ1 (˜ x) = x. We denote x(t) = φt (˜ x). We have : X X tk d k x |t=0 = tk Gk (t, x)|t=0 (63) x(t) = k k! dt k≥0 k≥0 where we denote Gk (t, x) =

1 dk x . So we have the following recurk! dtk

rence’s formulae : Gk+1 (t, x) =

∂Gk 1 ∂Gk 1 dGk (t, x) = ( + Xt . ) k + 1 dt k + 1 ∂t ∂x

(64)

Assume that we have G0 (t, x) = x, we thus compute the Gk up to a fixed order corresponding to the order of the truncation in the variables x, Y . Then we compute the function φt using (63) and the change of variables is given by x = x(1) = φ1 (˜ x). These transformations are computed using the computer algebra system TRIP, developed by J. Laskar and M. Gastineau.

articleMRL2.tex; 15/11/2001; 17:51; p.18

19 4.1.2.3. Properties of this transformation. new variables4 .

˜ x ˜ λ, We denote by Λ, ˜, y˜ the

Proposition. The transformation φ1 has the following form:  ˜ ) (Λ = Λ   P  j ˜ j 1≤j≤n (λj = λj + P∞ k=1 Pj,2k )1≤j≤n ∞  (x = x ˜ + j  k=1 Qj,2k+1 )1≤j≤n P∞  j (Yj = y˜j + k=1 Rj,2k+1 )1≤j≤n−1

(65)

where Pj,k , Qj,k and Rj,k are homogenous polynomials of degree k ˜ satisfying d’Alembert in (˜ x, y˜) with coefficients depending on the Λ relations and relations of parity in the variables y˜. The d’Alembert’s characteristic of each monomial of Pj,k , Qj,k and Rj,k is equal respectively to 0, 1, 1. Pj,k and Qj,k are even and Rj,k odd in the variables y˜. The Hamiltonian expressed in the new coordinates satisfies d’Alembert relations and parity in the variables y˜. Proof. As the function fy satisfies the d’Alembert relations and a relation of parity in the variables y, the vector field Xt and the new variables still satisfy d’Alembert relations and relations of parity in the inclination variables. The calculus required to check these facts are too long to be presented here but do not present major difficulties. In the coordinate system (Λ, λ, x, x ¯, Y, Y¯ ), we express the vector field Xt in the associated basis : Xt =

n X j=1

(Xt,Λj

∂ ∂ ∂ ∂ + Xt,λj + Xt,xj + Xt,¯xj )+ ∂Λj ∂λj ∂xj ∂x ¯j n−1 X j=1

(Xt,Yj

∂ ∂ + Xt,Y¯j ¯ ) ∂Yj ∂ Yj

(66)

Xt,. are the coordinates of Xt in the basis ∂/(∂.) of the tangent space. We have, by (59), Xt σt = Xt σ0 + tXt d(θ1 − θ0 )

(67)

The function fz does not depend of the variables λ and thus considering (58,60), Xt d(θ1 − θ0 ) ∈ vect(dΛ, dx, d¯ x, dY, dY¯ ) (68) ˜ = (λ ˜1, . . . , λ ˜ n ), x ˜ = (Λ ˜ 1, . . . , Λ ˜ n ), λ In the following, Λ ˜ = (˜ x1 , . . . , x ˜n ), y˜ = (˜ y1 , . . . , y˜n−1 ). 4

articleMRL2.tex; 15/11/2001; 17:51; p.19

20 where vect(dΛ, dx, d¯ x, dY, dY¯ ) is the linear subspace engendered by these vectors. And as Xt σ0 =

n X

Xt (Λj )dλj + Xt (λj )dΛj + Xt (xj )d¯ xj + Xt (¯ xj )dxj +

j=1

n−1 X

Xt (Yj )dY¯j + Xt (Y¯j )dYj ,

(69)

j=1

we have Xt σt =

n X

Xt (Λj )dλj + αt

(70)

j=1

where αt ∈ vect(dΛ, dx, d¯ x, dY, dY¯ ). But by (56), Xt σt = −∂θt /∂t = θ1 − θ0 . So Xt σt ∈ vect(dΛ, dx, d¯ x, dY, dY¯ ) and thus for 1 ≤ j ≤ n, we have Xt (Λj ) = 0. The transformation let unchanged the variables Λ. Moreover, Xt does not depend of the variables λ. Thus, it explains the fact that the polynomials Pj,k , Qj,k and Rj,k are independent of λ. From these three properties, we show that the Hamiltonian still satisfies the d’Alembert relations and that the parity in the new variables y˜ is verified. ¤ Remark 1. In the case where the inclinations are equal to zero (Y = 0), this transformation is the identity for the variables x, Λ, λ : the reduction in this case is already done. Remark 2. For three bodies, when we use this method, we find the transformation proposed in section 3. An expression of these transformations is given in the appendix for four bodies and truncated at degre 4.

4.1.2.4. The angular momentum ing property:

The reduced system has the follow-

Proposition. With the notations of the precedent paragraph, in the new system of coordinates, we have Cz =

n X j=1

˜j − Λ

n X j=1

|˜ xj | 2 −

n−1 X j=1

|˜ yj | 2 .

(71)

articleMRL2.tex; 15/11/2001; 17:51; p.20

21 Proof. The Hamilton flow at the time θ engendered by Cz (7) is the solution of the following system, for 1 ≤ j ≤ n :  x˙ j (θ) = ∂Cz /∂(−i¯ xj ) = −ixj (θ)    y˙ (θ) = ∂C /∂(−i¯ y j z j ) = −iyj (θ) (72) ˙ Λj (θ) = −∂Cz /∂(λj ) = 0    ˙ λj (θ) = ∂Cz /∂(Λj ) = 1

So the flow at the time θ is the transformation Ψθ defined, for 1 ≤ j ≤ n, by    −iθ  xj e xj  yj  Ψθ  e−iθ yj      (73)  Λj  →  Λj  λj λj + θ

While doing the partial reduction, we replace the variables yn , y¯n by the functions fy , f¯y of the variables Λ, x, y, x ¯ and y¯ (35). The angular momentum is then Cz =

n X j=1

Λj −

n X j=1

|xj |2 −

n−1 X j=1

We define the flow ψθ as the following map   −iθ  xj e xj  yk  ψθ  e−iθ yk     Λj  →  Λj λj λj + θ

|yj |2 − fy f¯y

   

(74)

(75)

for 1 ≤ j ≤ n and 1 ≤ k ≤ n − 1. fy satisfies d’Alembert relations: fy (e−iθ x, e−iθ y, Λ) = e−iθ fy (x, y, Λ) = e−iθ yn

(76)

The flow ψθ has thus the same action on the reduced system than the flow Ψθ defined by (73) and is then the flow generated by the integral Cz (74) in the reduced system. ψθ satisfies the following differential system :  for 1 ≤ j ≤ n, x˙ j (θ) = −ixj (θ)    for 1 ≤ j ≤ n − 1, y˙ (θ) = −iy (θ) j j (77) ˙ j (θ) = 0 for 1 ≤ j ≤ n, Λ    for 1 ≤ j ≤ n, λ˙ j (θ) = 1

That is X˙ = f (X) where the variables (xj , yk , Λj , λj ), for 1 ≤ j ≤ n and 1 ≤ k ≤ n−1, are denoted X. This new set of variables is not canonical, and, contrarily to (72), we have, for example, x˙ j (θ) 6= ∂Cz /∂(−i¯ xj ). To

articleMRL2.tex; 15/11/2001; 17:51; p.21

22 achieve the partial reduction, we construct a transformation φ = φ−1 1 ◦ ˜ ˜ P1 (45, 65) which defines new canonical variables (˜ x, y˜, Λ, λ) denoted ˜ φ conserves the d’Alembert relations, so X. φ(ψθ (X)) = ψθ (φ(X)).

(78)

To express the flow generated by Cz in this new set of variables, we have ˜˙ = Dφ(X)X˙ = Dφ(X)f (X) X (79) Derivating (78) with respect to θ, gives the identity Dφ(ψθ (X))f (ψθ (X)) = f (ψθ (φ(X))).

(80)

And for θ = 0, we have thus ˜ Dφ(X)f (X) = f (φ(X)) = f (X).

(81)

˜˙ = f (X). ˜ The flow of Cz is thus generated by the same Thus we have X ˜ As the variables X ˜ differential system (77) in the new coordinates X. are canonical, we have now the same expression of the flow that in (72) :  ¯˜j ) for 1 ≤ j ≤ n, x ˜˙ j (θ) = −i˜ xj (θ) = ∂Cz /∂(−ix     for 1 ≤ j ≤ n − 1, y˜˙ (θ) = −i˜ yj (θ) = ∂Cz /∂(−iy¯˜j ) j (82) ˙ ˜j ˜ j (θ) = 0 = −∂Cz /∂ λ  for 1 ≤ j ≤ n, Λ    ˜˙ j (θ) = 1 = ∂Cz /∂ Λ ˜j for 1 ≤ j ≤ n, λ And thus, doing an integration of the precedent relations, we have, in the new system of variables, Cz =

n X j=1

˜j − Λ

n X j=1

|˜ xj | 2 −

n−1 X j=1

|˜ yj | 2 + K

(83)

where K is a constant. As, by the transformation φ, the point (Λ, x = ˜ = 0), the ˜ x 0, y = 0, λ = 0) is changed in the point (Λ, ˜ = 0, y˜ = 0, λ constant K is equal to zero. ¤ The third integral of the angular momentum Cz has thus the same expression as in the original variables. It will allow us to define, in the next section, a new transformation, in the same way than for three bodies, in order to reduce the number of degrees of freedom of the system. 4.1.3. Origin point of the secular problem The origin point of the secular problem in the new variables x ˜1 = x ˜2 = ... = x ˜n = y˜1 = y˜2 = . . . = y˜n−1 = 0 is a fixed point of the system.

articleMRL2.tex; 15/11/2001; 17:51; p.22

23 So, from (43) and (65), we have x1 = x2 = . . . = xn = y1 = y2 = . . . = yn−1 = 0. And, as we have chosen that yn tends toward zero when the other y tend toward zero (35), we have also yn = 0. So the fixed point at the origin of the secular reduced system is the same as for the non reduced secular system. 4.2.

Total reduction

As in the 3-body case, the total reduction, using the third component of the angular momentum is possible. This second reduction, which can be done in a similar way even if the partial reduction is not achieved, has already been presented in the Andoyer variables in (Boigey, 1981). polar coordinates transformation : (˜ xj = q q We make the symplectic ˜ ˜j) ˜ j ei$ R yj = S˜j eiΩj )1≤j≤n−1 which gives: 1≤j≤n , (˜ Cz =

n X j=1

˜j − Λ

n X j=1

˜j − R

n−1 X

S˜j

(84)

j=1

As for three bodies (section 3), we use a linear change of variables in which Cz will be a new coordinate:  ˇ ˜ j )1≤j≤n ( Λj = Λ    (R ˇj = R ˜ j )1≤j≤n (85) (Sˇ = S˜j )1≤j≤n−2   Pn ˜ Pn ˜ Pn−1 ˜  ˇj Sn−1 = − j=1 Λj + j=1 Rj + j=1 Sj = −Cz By conjugation, the new angles are :  ˇj = λ ˜j + Ω ˜ n−1 )1≤j≤n (λ    ˜ n−1 )1≤j≤n ($ ˇj = $ ˜j − Ω ˇ =Ω ˜j − Ω ˜ n−1 )1≤j≤n−2  (Ω   ˇj ˜ (Ωn−1 = Ωn−1 )

(86)

q ˇj) ˇ j ei$ We then come back to rectangular variables : (ˇ xj = R 1≤j≤n , q ˇ (ˇ yj = Sˇj eiΩj )1≤j≤n−2 . So the total change of variables is:  ˜j = Λ ˇj  Λ    ˜ ˇ ˇ    λj = λj −ˇ Ωn−1 i Ω x ˜j = x ˇj e n−1 ˇ n−1  iΩ  y˜j = yˇj eq      y˜n−1 = −Cz + Pn

(87) P ˇ ˇ − Pn x ˇ¯j − n−2 ˇj yˇ¯j eiΩn−1 j=1 ˇj x j=1 y

j=1 Λj

articleMRL2.tex; 15/11/2001; 17:51; p.23

24 As for the three body’s reduction, the variable Ωn−1 does not appear in the Hamiltonian, due to the d’Alembert relations. The symplectic form becomes σ=

n X k=1

ˇ k ∧ dΛ ˇ k ∧ dx ˇ k − idx ¯ˇk ] − [dλ

n−2 X k=1

idˇ yk ∧ dy¯ˇk

(88)

And the total reduction associated with the angular momentum is thus achieved for n bodies. Remark 1. The main difference with the three-body case is that we do not have parity in the inclination variables yn−1 and thus we are compelled to expand the square root of the change of coordinates (87) as an infinite series. In this case, the new origin point of the system (ˇ x1 = x ˇ2 = . . . = x ˇn = yˇ1 = yˇ2 = . . . = yˇn−2 = 0) is not a fixed point of the secular problem. The study of the system after this reduction is not easy and it can be more valuable to stop after the partial reduction. Remark 2. In this method, we choose to eliminate the variable y˜n−1 , but there is no reason why we could not choose to eliminate any of the variable x ˜j , y˜j .

5. General remarks about these reductions 5.1. The method Our method to reduce the number of variables using integrals can be performed in a very general context and should be also of interest for other problems when we have an Hamiltonian system depending on the conjugated variables (p = (p1 , p2 , . . . , pn ), q = (q1 , q2 , . . . , qn )) with k integrals F1 (p, q), F2 (p, q), . . ., Fk (p, q). If we can expand these k integrals in infinite series and express k chosen variables as functions of the others ones, we replace these variables in the Hamiltonian and in the symplectic form which does not remain standard. Using a constructive version of the Darboux’s theorem, we then find a new transformation to restore the standard symplectic form. We also applied this method to a system, which is not so different from our present system : the secular problem at any order of the masses. TheP only change from the present case is that the two form σ becomes −i nk=1 (dxk ∧ d¯ xk + dyk ∧ d¯ yk ) as the Λ are now parameters of the problem.

articleMRL2.tex; 15/11/2001; 17:51; p.24

25 5.2. Constructibility of a Birkhoff normal form The construction of a Birkhoff normal form (Birkhoff, 1927) is a classical tool to study the dynamics of the secular problem (for example to apply a KAM theorem (Arnold, 1963, Robutel, 1995), to study the secular frequencies of a planetary problem (Brumberg, 1980, Laskar, 1984) or for the calculus of the asteroid proper elements (Milani and Knezevic, 1990)). We start with a Hamiltonian of the form H=

2n X j=1

νj zj z¯j +

X

Hk (z1 , z2 , . . . , z2n , z¯1 , z¯2 , . . . , z¯2n )

(89)

k≥3

where zj , −i¯ zj are symplectically conjugated and Hk is a homogeneous polynomial of degree k in z, z¯. In our case, H is the Hamiltonian of the secular problem at order one with respect to the masses, calculated as the average of the Hamiltonian H1 over the fast angles λ. We construct then formally, degree by degree, a canonical transformation (z → z˜) so ˜ depends only on the actions J˜ = z˜j z¯˜j . that the new Hamiltonian H During this construction, using any algorithm (see Deprit, 1969), diviP −1 will appear, where k , k , . . . , k sors of the form ( 2n k ν j j) 1 2 2n ∈ Z. j=1 If the νj are rationally dependent, some divisors are equal to zero and can prevent the construction of the normal form. In our case, there exist indeed two such resonance relations that hold for all values of the parameters (masses, semi-major axis and constant of gravitation). The first one, well known, is the occurence of a zero secular frequency. This null eigenvalue is directly linked to the invariance of the direction of the angular momentum. Indeed, in the linear problem, at order 1 in x, y, C1 is reduced to C11 : C11

=

n X j=1

yj

p

Λj

(90)

And thus C11 is an eigenvector for the linear system, of eigenvalue equal to zero. The linear problem has thus one of its secular frequencies equal to zero and we fix ν2n = 0. The second relation, that we will call Herman’s resonance, is less known: we have ν1 + ν2 + . . . + ν2n = 0. Abdullah and Albouy (2001) give a demonstration of this relation in a general situation. In fact, this relation does not come from an integral of the problem and is satisfied only for the secular problem at order one with respect to the masses and can be easily checked using the expressions of H2 , given in the appendix C. So, there is a risk that some zero divisors will prevent the construction of a non-resonant normal form at all order and for all values of

articleMRL2.tex; 15/11/2001; 17:51; p.25

26 ¯

kj kj in the well-known homological the parameters. A term Π2n j=1 zj z¯j equation (present in all P schemes of construction of a Birkhoff normal 2n ¯ form) gives the divisor j=1 νj (kj − kj ). And thus, a term such as 2n−1 2n−1 (Πj=1 zj )¯ z2n , which satisfy the d’Alembert relation, provides a zero divisor when we use the precedent relations. So, in general and without a precise study, the construction of a Birkhoff normal form at all degrees is not granted. In fact, we can prove, with a more precise study, that for three bodies these resonances will actually prevent the construction of the Birkhoff normal form at degree ten and more in eccentricities and inclinations (Malige, 2001). We can show that, after the partial reduction, no problem due to these resonance relations will occur during the construction. Indeed, the zero frequency, coming from the invariance of C1 , is not present in the partially reduced system. In the homological equation, a term ¯ P2n−1 P2n−1 2n−1 kj ¯k Πj=1 z˜j z˜j j give the divisor : j=1 νj (kj − k¯j ). As j=1 νj = 0, we still have a possible zero divisor when there exists p ∈ Z such as for all j, 1 ≤ j ≤ 2n − 1, kj − k¯j = p. But, as the Hamiltonian still satisfies the P2n−1 d’Alembert relations after the partial reduction, j=1 kj − k¯j = 0 and then p = 0. The only resonant terms, for all values of the parameters, k 2n−1 are thus the terms Πj=1 (˜ zj z¯˜j ) j that are kept in the normal form. The partial reduction thus enables us to avoid all problems of resonance. If the secular problem is computed at a higher order of the masses, the sum of the frequencies is no longer equal to zero but, in certain cases, remains small, compared to the frequencies. So we can have divisors which are not equal to zero but close to zero. The partial reduction prevents such terms to appear. For our Solar System, the change of the sum of the frequencies of the linear secular system between order one and two is quite important (1.5 arcsec/yr), of about the same amplitude as the secular frequencies themselves (see Laskar, 1985).

5.3. The invariant plane The invariant plane, orthogonal to the angular momentum, plays a central role in the partial reduction. Indeed, if we are in a different reference plane, the integral C1 is no longer equal to zero. In the expression of yn (35), we have s n−1 X |yj |2 2 (91) α = C1 − yj Λj − |xj | − 2 j=1 with C1 6= 0. Thus, we replace yn in the Hamiltonian by its expression in (35), the new Hamiltonian no longer satisfies the d’Alembert relations and the parity in the variables y. The situation is more difficult to

articleMRL2.tex; 15/11/2001; 17:51; p.26

27 understand : if the inclinations are set to zero at the beginning of the movement they do not remain equal to zero as in the situation of the invariant plane (plane problem of n bodies). The point x ˜1 = x ˜2 = . . . = x ˜n = y˜1 = y˜2 = . . . = y˜n−1 = 0 is no longer a fixed point of the secular system. Besides, the formal solutions of the problem in the invariant plane reference frame are infinite quasi-periodic series depending of 3n − 1 non null frequencies ν1 , ν2 , . . . , ν3n−1 : if Z is one of the complex polar ˜ j ), x coordinates (Λj exp(iλ ˜j , y˜j ), we have X Z(t) = ak exp(i(k1 ν1 + . . . + k3n−1 ν3n−1 )t), (92) k∈Z3n−1

And these infinite series satisfy d’Alembert relations : ak = 0 if

3N −1 X j=1

kj 6= σ(Z)

(93)

˜ j )) = 0 and σ(˜ where σ(Λj exp(iλ xj ) = σ(˜ yj ) = 1. These relations on the kj prevent that the expressions of x ˜j or y˜j contain constant terms. A change of reference plane will transform the Euclidean coordinates system by an Euler’s transformation R. It can be written as a product of three basic rotations : R = R3 (θ3 )R1 (θ2 )R3 (θ1 ) where θ2 represent the angle between the two planes and the rotations are defined by     1 0 0 cos(θ) sin(θ) 0 R1 (θ) =  0 cos(θ) sin(θ)  and R3 (θ) =  − sin(θ) cos(θ) 0  0 − sin(θ) cos(θ) 0 0 1 (94) In order to compute the new elliptical elements Ij , Ωj in the new reference frame, we define zj =

1 ((Cx )j + i(Cx )j ) = sin Ij sin Ωj − i sin Ij cos Ωj . kCj k

(95)

We denote Rk (zj ) the new value of zj by any rotation Rk . We have : R3 (θ)(zj ) = eiθ zj and

q R1 (θ)(zj ) = zj cos (θ/2) + z¯j sin (θ/2) + i sin θ 1 − |zj |2 2

2

(96) (97)

The new value of zj , Rzj is, in the new reference frame,

Rzj = zj cos2 (θ2 /2)ei(θ3 +θ1 ) + z¯j sin2 (θ2 /2)ei(θ3 −θ1 )

articleMRL2.tex; 15/11/2001; 17:51; p.27

28 −i sin θ2 ei(θ3 +θ1 )

q

1 − |zj |2

(98)

q And, owing to the term 1 − |zj |2 in (98), in the new reference frame, constants terms appear in the values of the inclinations and nodes’s angles and thus in the expression of the y˜ and the x, y, as it was observed in secular theories conducted in the J2000 reference frame (see for example Laskar, 1987).

Appendix A

We present here two different ways to achieve the linear transformation that make the symplectic form standard at order zero (section 4.1.1.). The first possible orthogonal transformation is the orthogonal symmetry with respect to the hyperplane orthogonal to v = V/ kVk − en−1 , that can be computed using (44) as :

√ Dn−1 + Λn−1 Q = In−1 − 2 Dn−1 Dn−2

√ Dn−1 + Λn−1 + 2 Dn−2





√ Λ Λ1 Λ2 1  √Λ1 Λ2 Λ2   .. ..  √ . √ . Λ1 Λn−1 Λ2 Λn−1

√ . . . √Λ1 Λn−1 . . . Λ2 Λn−1 .. .. . . ... Λn−1

√ Λ1 ... 0  .. .. . .  . .  √ .  0 ... Λn−2 √ √ √ 0 Λ1 . . . Λn−2 2 Λn−1 − Dn−1 0 .. .

    

    

qP

k where Dk = i=1 Λi and In is the (n − 1) × (n − 1) identity matrix. In this linear transformation, the variable n − 1 plays a special role but there are symmetrical relations between the n − 2 others. The second possible orthogonal transformation is obtained through a Schmidt’s orthogonalisation where the matrix Q becomes :

articleMRL2.tex; 15/11/2001; 17:51; p.28

29 √ √ √ Λ1 Λ2 Λ1 Λ3 Λ1 Λ4 − − −  D D D D D D 1 2 2 3  √ √3 4  Λ2 Λ3 Λ2 Λ4 D1  − −  D2 D2 D3 D D  √3 4   D Λ3 Λ4 2  0 −  D D 3 3 D4 Q=  D3  0 0  D4   .. .. ..   . . .   0 0 0 

p

Λ1 Λn−1 D n−2 Dn−1 p Λ2 Λn−1 ... − D n−2 Dn−1 p Λ3 Λn−1 ... − Dn−2 Dn−1 .. ... . ... −

..

.

...

.. . Dn−2 Dn−1





Λ1

Dn−1 √ Λ2 − Dn−1 √ Λ3 − Dn−1 .. . .. p. Λn−1 − Dn−1

                    

Appendix B We present, finally, the partial reduction in the case of 4 bodies, using the second linear transformation (Schmidt’s orthogonalisation). The effective computation of this transformation takes about one minute using the computer algebra system TRIP (which methods are presented in Laskar, 1989b) on a work station. The expressions are truncated at the degree 4.

x1 = x ˜1 √ −1 √ √ −1 −1 ˜1 y˜2 y¯˜1 Λ1 + 14 x Λ Λ D √ −1 √ 2 √ 3−1 3−1 ˜1 y˜1 y¯˜2 Λ1 Λ2 Λ3 D3 − 14 x + ◦ (˜ x, y˜)4

x2 = x ˜2 √ √ −1 √ −1 −1 Λ D ˜2 y˜2 y¯˜1 Λ1 Λ2 − 14 x √ √ −1 √ 3−1 3−1 Λ3 D3 + 14 x ˜2 y˜1 y¯˜2 Λ1 Λ2 + ◦ (˜ x, y˜)4

x3 = x ˜3 +◦ (˜ x, y˜)4

articleMRL2.tex; 15/11/2001; 17:51; p.29

30

˜ 1 + i( λ1 = λ − 12 + 18 − 18 + 18 − 18 1 + 16 1 + 2 − 18 + 18 − 18 + 18 1 + 16 1 − 16 − 81 5 + 16 1 − 4 1 + 16 1 − 16 1 − 16 1 + 16 1 + 8 5 − 16 1 + 4 1 − 16 1 + 4 − 14 − 18 + 18 + 18 − 18 + 18 + 18 + 14 + 18 − 18 − 18 − 14 − 18 + ◦

√ −1 √ √ Λ2 Λ3 D2−2 D3−1 y˜2 y¯˜1 Λ1 √ √ √ −1 −3 −1 y˜22 y¯˜1 y¯˜2 Λ1 Λ Λ D √ √ 2 √ 3 −53 −1 y˜22 y¯˜1 y¯˜2 Λ1 Λ Λ D √ −1 √ 2 √ 3 3−2 −3 y˜22 y¯˜1 y¯˜2 Λ1 Λ Λ D D √ −1 √ 2 √ 33 2−2 3−5 Λ Λ3 D2 D3 y˜22 y¯˜1 y¯˜2 Λ1 √ −2 −2 2 2 ¯2 y˜2 y˜1 Λ1 D3 √ −1 √ √ y˜1 y¯˜2 Λ1 Λ Λ D2−2 D3−1 √ −1 √2 √3 −1 2 y˜1 y˜2 y¯˜2 Λ1 Λ Λ D−3 √ −1 √ 2 √ 3 −53 2 Λ2 Λ3 D3 y˜1 y˜2 y¯˜2 Λ1 √ 2 √ −1 √ ¯ Λ2 Λ3 D2−2 D3−3 y˜1 y˜2 y˜2 Λ1 √ 3 √ √ −1 2 Λ2 Λ3 D2−2 D3−5 y˜1 y˜2 y¯˜2 Λ1 √ √ −3 5 √ −1 2 y˜1 y˜2 y¯˜1 Λ1 Λ2 Λ3 D2−4 D3−1 √ √ −1 −1 √ 2 y˜1 y˜2 y¯˜1 Λ1 Λ2 Λ D−3 √ −1 √ √ −13 3−3 2 Λ2 Λ3 D3 y˜1 y˜2 y¯˜1 Λ1 √ −1 2 √ −1 √ Λ2 Λ3 D2−2 D3−1 y˜1 y˜2 y¯˜1 Λ1 √ √ √ −1 2 Λ2 Λ3 D2−2 D3−3 y˜1 y˜2 y¯˜1 Λ1 √ √ −1 3 √ −1 2 y˜1 y˜2 y¯˜1 Λ1 Λ2 Λ3 D2−4 D3−1 2 √ −2 y˜12 y¯˜2 Λ1 D3−2 √ −3 √ 5 √ −1 −4 −1 Λ Λ D2 D3 y˜12 y¯˜1 y¯˜2 Λ1 √ −1 √ 2−1 √ 3 Λ Λ D−3 y˜12 y¯˜1 y¯˜2 Λ1 √ −1 √ 2 √ −13 3−3 2¯ ¯ Λ Λ D y˜1 y˜1 y˜2 Λ1 √ −1 √ 2 √ 3−1 3−2 −1 2¯ ¯ y˜1 y˜1 y˜2 Λ1 Λ Λ D D √ −1 √ 2 √ 3 −22 −33 y˜12 y¯˜1 y¯˜2 Λ1 Λ Λ D D √ −1 √ 23 √ 3−1 2 −4 3 −1 Λ 2 Λ 3 D2 D3 y˜12 y¯˜1 y¯˜2 Λ1 √ √ −1 −3 √ ¯˜3 y˜2 y¯˜1 Λ−1 Λ Λ D x ˜3 x 1 √ −1 √ 2 √ 3−1 3−3 ¯ ¯ Λ Λ D3 x ˜3 x ˜3 y˜1 y˜2 Λ1 √ √ 2−1 √ 3 ¯˜2 y˜2 y¯˜1 Λ−1 x ˜2 x Λ Λ D−3 1 √ −1 √ 2 √ −13 3−3 ¯˜2 y˜2 y¯˜1 Λ1 x ˜2 x Λ Λ D3 √ √ 2−1 √ 3 ¯˜2 y˜1 y¯˜2 Λ−1 x ˜2 x Λ Λ D−3 √ 1−1 √ 2 √ −13 3−3 ¯˜2 y˜1 y¯˜2 Λ1 Λ Λ D x ˜2 x √ 23 √ 3−1 3−3 √ ¯˜1 y˜2 y¯˜1 Λ−3 Λ Λ D x ˜1 x 1 √ √ 23 √ 3 −23 −3 ¯˜1 y˜2 y¯˜1 Λ−3 x ˜1 x Λ Λ D D √ 1 √ 2 √ 3−1 2 −3 3 ¯˜1 y˜2 y¯˜1 Λ−1 x ˜1 x Λ Λ D √ 1−1 √ 2 √ 3 −23 −3 ¯ ¯ Λ Λ D D x ˜1 x ˜1 y˜2 y˜1 Λ1 √ 23 √ 3−1 2 −3 3 √ ¯˜1 y˜1 y¯˜2 Λ−3 Λ Λ D x ˜1 x 1 √ 23 √ 3 −23 −3 √ ¯˜1 y˜1 y¯˜2 Λ−3 Λ Λ D D x ˜1 x √ 1 √ 2 √ 3−1 2 −3 3 ¯˜1 y˜1 y¯˜2 Λ−1 x ˜1 x Λ Λ D √ 1−1 √ 2 √ 3 −23 −3 ¯ ¯ x ˜1 x ˜1 y˜1 y˜2 Λ1 Λ2 Λ3 D2 D3 ) (˜ x, y˜)4

articleMRL2.tex; 15/11/2001; 17:51; p.30

31

˜ 2 + i( λ2 = λ + 21 − 81 + 81 − 81 + 81 1 + 16 1 − 2 + 81 − 81 + 81 − 81 1 − 16 1 − 16 1 − 16 1 + 8 + 81 1 − 16 1 + 16 1 + 16 1 + 16 − 81 − 81 − 41 + 41 + 81 − 83 + 41 − 81 − 81 + 83 − 41 + 81 − 81 + 81 + 81 + 81 − 81 − 81 + ◦

√ √ −1 √ Λ3 D2−2 D3−1 y˜2 y¯˜1 Λ1 Λ2 √ √ √ −1 −3 −1 y˜22 y¯˜1 y¯˜2 Λ1 Λ2 Λ D √ √ √ 3 −53 −1 y˜22 y¯˜1 y¯˜2 Λ1 Λ2 Λ D √ √ −1 √ 3 3−2 −3 y˜22 y¯˜1 y¯˜2 Λ1 Λ2 Λ D D √ √ −1 √ 33 2−2 3−5 Λ3 D2 D3 y˜22 y¯˜1 y¯˜2 Λ1 Λ2 √ −2 −2 2 ¯2 y˜2 y˜1 Λ2 D3 √ √ −1 √ y˜1 y¯˜2 Λ1 Λ2 Λ D2−2 D3−1 √ √ −1 √3 −1 2 y˜1 y˜2 y¯˜2 Λ1 Λ2 Λ D−3 √ √ −1 √ 3 −53 2 Λ D y˜1 y˜2 y¯˜2 Λ1 Λ2 √ −1 √ 3 3−2 −3 2√ ¯ Λ D D y˜1 y˜2 y˜2 Λ1 Λ2 √ −1 √ 33 2−2 3−5 2√ Λ3 D2 D3 y˜1 y˜2 y¯˜2 Λ1 Λ2 2 √ −1 √ −3 √ −1 4 −3 y˜1 y˜2 y¯˜1 Λ1 Λ2 Λ3 D2 D3 2 √ −1 √ −3 √ y˜1 y˜2 y¯˜1 Λ1 Λ2 Λ D2 D−3 √ −1 √ −1 √ 3−1 2 2 3 −3 2 Λ2 Λ3 D2 D3 y˜1 y˜2 y¯˜1 Λ1 2 √ −1 √ −1 √ Λ2 Λ D−3 y˜1 y˜2 y¯˜1 Λ1 √ −1 √ √ −13 3−3 2 Λ2 Λ3 D3 y˜1 y˜2 y¯˜1 Λ1 2 √ −2 y˜12 y¯˜2 Λ2 D3−2 √ −1 √ −3 √ −1 4 −3 y˜12 y¯˜1 y¯˜2 Λ1 Λ Λ D D √ −1 √ 2−3 √ 3 2 2 −33 Λ2 Λ D D y˜12 y¯˜1 y¯˜2 Λ1 √ −1 √ −1 √ 3−1 2 2 3 −3 Λ2 Λ D D y˜12 y¯˜1 y¯˜2 Λ1 √ −1 √ −1 √ 3 −32 3 Λ2 Λ D y˜12 y¯˜1 y¯˜2 Λ1 √ −1 √ √ −13 3−3 2¯ ¯ y˜1 y˜1 y˜2 Λ1 Λ Λ D √ √ 2 √3 −13 −3 ¯˜3 y˜2 y¯˜1 Λ1 Λ−1 x ˜3 x Λ D √ √ 2 √ 3−1 3−3 ¯˜3 y˜1 y¯˜2 Λ1 Λ−1 Λ D x ˜3 x 2 √ 3−1 3−3 √ √ ¯˜2 y˜2 y¯˜1 Λ1 Λ−1 Λ D x ˜2 x √ √ 2−1 √ 3−1 3−2 −1 ¯ ¯ Λ D D x ˜2 x ˜2 y˜2 y˜1 Λ1 Λ2 √ √ √ 3 −22 −33 ¯˜2 y˜2 y¯˜1 Λ1 Λ−1 x ˜2 x Λ D D 2 √ √ √ 3−1 2 −2 3 −1 ¯˜2 y˜2 y¯˜1 Λ31 Λ−3 x ˜2 x Λ D D √ √ 2 √ 3−1 2−3 3 ¯˜2 y˜1 y¯˜2 Λ1 Λ−1 x ˜2 x Λ D √ √ 2−1 √ 3−1 3−2 −1 ¯ ¯ Λ D D x ˜2 x ˜2 y˜1 y˜2 Λ1 Λ2 √ 3 −22 −33 √ √ ¯˜2 y˜1 y¯˜2 Λ1 Λ−1 Λ D D x ˜2 x 2 √ √ √ 3−1 2 −2 3 −1 ¯˜2 y˜1 y¯˜2 Λ31 Λ−3 x ˜2 x Λ D D √ √ 2 −1 √ 3 −1 2 2 3−3 ¯˜1 y˜2 y¯˜1 Λ−1 x ˜1 x Λ Λ D D √ 1−1 √ 2−1 √ 3 −32 3 ¯ ¯ Λ Λ D x ˜1 x ˜1 y˜2 y˜1 Λ1 √ 2 √ −13 3−3 √ ¯˜1 y˜2 y¯˜1 Λ−1 Λ2 Λ3 D3 x ˜1 x 1 √ −1 √ −1 2 −3 √ ¯˜1 y˜1 y¯˜2 Λ−1 Λ Λ D D x ˜1 x √ 1−1 √ 2−1 √ 3 −32 3 ¯˜1 y˜1 y¯˜2 Λ1 x ˜1 x Λ2 Λ D √ −1 √ √ −13 3−3 ¯ ¯ x ˜1 x ˜1 y˜1 y˜2 Λ1 Λ2 Λ3 D3 ) (˜ x, y˜)4

articleMRL2.tex; 15/11/2001; 17:51; p.31

32

˜ 3 + i( λ3 = λ 1 − 16 1 + 8 1 + 16 1 − 8 − 18 + 18 + 18 − 18 + 18 − 18 + ◦

2 √ −1 √ −1 √ −3 4 −3 Λ2 Λ D2 D3 y˜1 y˜2 y¯˜1 Λ1 √ −33 2 √ −1 √ Λ2 Λ3 D22 D3−3 y˜1 y˜2 y¯˜1 Λ1 √ −1 √ −1 √ −3 4 −3 Λ Λ D D y˜12 y¯˜1 y¯˜2 Λ1 √ −1 √ 2 √ −33 2 2 −33 y˜12 y¯˜1 y¯˜2 Λ1 Λ2 Λ3 D2 D3 √ √ √ −3 −1 ¯˜2 y˜2 y¯˜1 Λ1 Λ−1 x ˜2 x Λ D √ √ 2−1 √ 3−1 3−3 ¯˜2 y˜2 y¯˜1 Λ1 Λ2 Λ D x ˜2 x √ 3−3 3−1 √ √ ¯˜2 y˜1 y¯˜2 Λ1 Λ−1 Λ D x ˜2 x √ √ 2 √ 3−1 3−3 ¯˜2 y˜1 y¯˜2 Λ1 Λ−1 Λ D x ˜2 x √ −1 √ 2 √ 3−3 32 −3 ¯ ¯ x ˜1 x ˜1 y˜2 y˜1 Λ1 Λ Λ D D √ √ 2 √ 3−3 22 3−3 ¯˜1 y˜1 y¯˜2 Λ−1 x ˜1 x Λ2 Λ3 D2 D3 ) 1 (˜ x, y˜)4

y1 = √ √ − 1 y˜2 Λ1 Λ3 D2−1 D3−1 √ √ −1 + 41 y˜22 y¯˜2 √Λ1 √Λ3 D2−1 D3−1 2¯ 1 − 2 y˜2 y˜2 √Λ1 Λ3 D2−1 D3−3 y¯˜1 Λ2 D2−1 D3−2 − 41 y˜22√ − 1 y˜1 Λ2 D2−1 √ −1 √ −1 3 −3 Λ D2 D3 + 14 y˜1 y˜2 y¯˜1 Λ1 √ −1 √ 3 1 ¯ Λ D D−3 − 4 y˜1 y˜2 y˜1 Λ1 √ −1 √ 32 √2 −13 1 ¯ Λ Λ D2 D3−3 − 2 y˜1 y˜2 y˜1 Λ1 √ −1 √ −12 −13 2¯ 1 Λ D D + 8 y˜1 y˜2 Λ1 √ −1 √ 3 −12 −13 − 18 y˜12 y¯˜2 Λ1 Λ D D √ √ −13 2−1 3−1 2¯ 1 − 4 y˜1 y˜2 Λ1 Λ3 D2 D3 √ −1 − 18 y˜12 y¯˜1 Λ2 D2−1 √ −1 √ 2 −1 −2 Λ D D3 + 18 y˜12 y¯˜1 Λ2 √ 2 √ −13 2 + 14 y˜12 y¯˜1 Λ1 Λ2 D2−1 D3−2 √ √ −1 −1 ¯˜3 y˜2 Λ1 Λ−1 + 12 x ˜3 x 2 D3 √ √ 3 D −1 −3 1 ¯ ˜3 x ˜3 y˜2 Λ1 Λ3 D2 D3 − 2 x √ √ −1 −1 ¯˜2 y˜2 Λ1 Λ−1 + 41 x ˜2 x 2 D3 √ √ 3 D −1 −3 1 ¯ ˜2 x − 2 x ˜2 y˜2 Λ1 Λ3 D2 D3 √ √ ¯˜2 y˜1 Λ21 Λ−1 ˜2 x D2−1 D3−2 + 41 x √ −1 √ 2 1 ¯˜1 y˜2 Λ1 Λ3 D2 D3−3 − 2 x ˜1 x √ √ −1 2 √ −1 ¯˜1 y˜2 Λ1 Λ2 Λ3 D2 D3−3 − 41 x ˜1 x √ √ −1 2√ ¯˜1 y˜2 Λ1 Λ2 Λ3 D2−1 D3−3 + 41 x ˜1 x √ ¯˜1 y˜1 Λ2 D2−1 D3−2 ˜1 x − 41 x + ◦ (˜ x, y˜)4

articleMRL2.tex; 15/11/2001; 17:51; p.32

33 y2 = √ √ − 1 y˜2 Λ2 Λ3 D2−1 D3−1 √ √ −1 + 41 y˜22 y¯˜2 √Λ2 √Λ3 D2 D3−3 2 − 41 y˜2 y¯˜2 √Λ2 Λ3 D2−1 D3−3 y¯˜1 Λ1 D2−1 D3−2 + 41 y˜22√ + 1 y˜1 Λ1 D2−1 √ −1 √ −1 3 −3 Λ D2 D3 − 14 y˜1 y˜2 y¯˜1 Λ2 √ −1 √ 3 Λ D D−3 − 41 y˜1 y˜2 y¯˜1 Λ2 √ √ −13 2 3−3 1 ¯ + 2 y˜1 y˜2 y˜1 Λ2 Λ3 D2 D3 √ −1 √ −1 −1 Λ D D − 18 y˜12 y¯˜2 Λ2 √ −1 √ 3 −12 −13 2¯ 1 + 8 y˜1 y˜2 Λ2 Λ D D √ 2 √ −13 √2 −1 3 −1 −1 Λ3 D2 D3 + 14 y˜12 y¯˜2 Λ1 Λ2 √ −1 − 18 y˜12 y¯˜1 Λ1 D2−1 √ −1 √ 2 −1 −2 Λ3 D2 D3 + 18 y˜12 y¯˜1 √Λ1 + 14 y˜12 y¯˜1 Λ1 D2−1 D3−2 √ √ −3 ¯˜3 y˜2 Λ2 Λ−1 ˜3 x + 12 x 3 D2 D3 √ √ −1 ¯˜2 y˜2 Λ2 Λ D−1 D−1 − 21 x ˜2 x √ −1 √ 33 2−1 3−3 1 ¯ Λ D D + 2 x ˜2 y˜2 Λ2 ˜2 x √ √ 3 √2 −1 3 −1 −1 ¯˜2 y˜2 Λ21 Λ−1 − 41 x ˜2 x Λ D D √ √ 2 √ 3 −12 −33 ¯˜2 y˜2 Λ21 Λ−1 ˜2 x Λ3 D2 D3 + 21 x 2 √ ¯˜2 y˜1 Λ1 D2−1 D3−2 ˜2 x + 41 x √ √ −3 ¯˜1 y˜2 Λ2 Λ−1 + 41 x ˜1 x 2 D3 √ √ 3 D −1 −3 1 ¯ ˜1 x ˜1 y˜2 Λ2 Λ3 D2 D3 − 4 x √ √ 2 −1 −2 ¯˜1 y˜1 Λ−1 ˜1 x Λ2 D2 D3 − 41 x 1 + ◦ (˜ x, y˜)4 y3 = + 1 y˜2 D2−1 D3 √ 2 − 1 y˜2 Λ3 D2−1 D3−1 + 14 y˜22 y¯˜2 D2−1 D3−1 √ 2 − 12 y˜22 y¯˜2 Λ3 D2−1 D3−3 √ 2 − 12 y˜1 y˜2 y¯˜1 Λ3 D2−1 D3−3 −1 −1 2¯ 1 − 4 y˜1 y˜2 D2 D3 √ −1 √ −1 √ −1 −1 2 Λ Λ D D − 18 y˜12 y¯˜1 Λ1 √ −1 √ 2−1 √ 33 −12 −23 + 81 y˜12 y¯˜1 Λ1 Λ2 Λ D D √ √ −1 √ −13 2−1 3 Λ3 D2 + 14 y˜12 y¯˜1 Λ1 Λ2 √ √ −1 √ Λ3 D2−1 D3−2 + 41 y˜12 y¯˜1 Λ1 Λ2 −3 1 ¯˜3 y˜2 D2 D3 ˜3 x + 2 x √ ¯˜2 y˜2 Λ23 D2−1 D3−3 ˜2 x − 12 x √ −1 −1 √ √ ¯˜2 y˜1 Λ1 Λ−1 Λ D + 41 x ˜2 x 2 √ √ 3 −12 −2 √ −1 ¯˜2 y˜1 Λ1 Λ2 Λ3 D2 D3 + 41 x ˜2 x √ ¯˜1 y˜2 Λ23 D2−1 D3−3 − 21 x ˜1 x √ √ √ −1 −1 ¯˜1 y˜1 Λ−1 − 41 x ˜1 x Λ Λ D 1 √ −1 √ 2 √ 3 −12 −2 1 ¯ ˜1 x ˜1 y˜1 Λ1 Λ2 Λ3 D2 D3 − 4 x + ◦ (˜ x, y˜)4

articleMRL2.tex; 15/11/2001; 17:51; p.33

34 Appendix C We present, in this appendix, the analytical expressions of the secular Hamiltonian at order one with respect to the masses for the threebody planetary problem. The Hamiltonian is expanded in the Poincar´e’s variables, truncated at order 4 in eccentricities and inclinations in the non-reduced (NR), partially reduced (PR) and totally reduced (TR) cases. The expressions depend on the following parameters and variables: α = a1 /a2 , ratio of the semi major axis Λ1 k= Λ2 s 2 (j = 1 or 2) Xj = xj Λj s 1 Y j = yj (j = 1 or 2) 2Λj s 1 Y =y 2(Λ1 + Λ2 ) s s 1 Λ1 + Λ2 − C z D=d = 2(Λ1 + Λ2 ) 2(Λ1 + Λ2 ) 2 2 K = −µ1 µ2 β1 β2 m2 /Λ2 m0 and of the coefficients Cj (α) which are also given in this appendix. The xj , yj are the Poincar´e’s variables, y the variable of the partially reduced system (section 3.1) and d the parameter appearing in the totally reduced system (remark 4 in the section 3.2). For the non reduced Hamiltonian, we take the notations of (Laskar and Robutel, 1995):

H2, +

NR

= K( ¯2 + X ¯ 1 X2 ) C2 (α) (X1 X 1 ¯1 + X ¯ 2 X2 ) + 2(Y1 Y¯2 + Y¯1 Y2 − Y1 Y¯1 − Y¯2 Y2 ))) C3 (α) ( (X1 X 2

articleMRL2.tex; 15/11/2001; 17:51; p.34

35

H4, + + + +

+ + + + + + + + + +

NR

= K( ¯ 2 Y1 Y¯2 + X ¯ 1 X2 Y¯1 Y2 ) C4 (α) (X1 X 2 ¯ X2 + X ¯1X ¯2X 2) C5 (α) (X1 X 2 2 ¯1X ¯ 2 + X1 X ¯ 2 X2 ) C6 (α) (X12 X 1 ¯ 2 Y1 Y¯1 + X ¯ 1 X2 Y1 Y¯1 + X1 X ¯ 2 Y¯2 Y2 + X ¯ 1 X2 Y¯2 Y2 ) C7 (α) (X1 X ¯ ¯ ¯ ¯ ¯ ¯ C8 (α) (2(X1 X2 Y1 Y2 + X1 X2 Y1 Y2 − X1 X2 Y1 Y2 ¯ 2 Y¯1 Y2 ) − X ¯1X ¯ 2 Y 2 − X1 X2 Y¯ 2 −X1 X 1 1 ¯1X ¯2Y 2) −X1 X2 Y¯22 − X 2 ¯ 2 Y 2 + X 2 Y¯ 2 + X 2 Y¯ 2 + X ¯ 2Y 2 C9 (α) (X 2 1 2 1 2 2 2 2 2 2 ¯ ¯ ¯ −2X2 Y1 Y2 − 2X2 Y1 Y2 ) ¯2 + X ¯ 2X 2) C10 (α) (X12 X 2 1 2 ¯ 1 Y1 Y¯1 + X ¯ 2 X2 Y¯2 Y2 ) C11 (α) (X1 X 2 ¯ ¯ ¯ C12 (α) (Y1 Y1 Y2 + Y1 Y12 Y2 + Y1 Y¯22 Y2 + Y¯1 Y¯2 Y22 ) ¯ 2) C13 (α) (X12 X 1 2 C14 (α) (Y1 Y¯12 + Y¯22 Y22 + Y12 Y¯22 + Y¯12 Y22 ) ¯ 1 Y1 Y¯2 + X ¯ 2 X2 Y1 Y¯2 + X1 X ¯ 1 Y¯1 Y2 + X ¯ 2 X2 Y¯1 Y2 ) C15 (α) (X1 X 2 2 ¯ C16 (α) (X2 X2 ) ¯ 2 Y 2 + X 2 Y¯ 2 + X 2 Y¯ 2 + X ¯ 2Y 2 C17 (α) (X 1 1 1 1 1 2 1 2 2 2 ¯ Y1 Y2 ) −2X1 Y¯1 Y¯2 − 2X 1 1 ¯1X ¯ 2 X2 + 4Y1 Y¯1 Y¯2 Y2 C18 (α) ( X1 X 4 ¯ 2 X2 Y1 Y¯1 − X1 X ¯ 1 Y¯2 Y2 ) ) −X

The coefficients Ck are defined by the following expressions, depend(k) ing of the well-known Laplace coefficients bs (Laskar and Robutel, 1995).

C1 (α) = C2 (α) = C3 (α) =

1 2 3 8 1 4

(0)

b1/2 (0)

αb3/2 − (

1 4

+

1 4

(1)

α2 )b3/2

(1)

αb3/2

C4 (α) = (−

15 4

α−

15 4

(0)

α3 )b5/2 + (

3 2

+

27 8

α2 +

3 2

(1)

α4 )b5/2

articleMRL2.tex; 15/11/2001; 17:51; p.35

36 C5 (α) C6 (α) C7 (α) C8 (α) C9 (α) C10 (α) C11 (α) C12 (α) C13 (α) C14 (α) C15 (α) C16 (α) C17 (α) C18 (α)

15

15

(0)

α+

α3 )b5/2 + (

3

9



α2 −

3

(1)

α4 )b5/2 64 64 32 64 32 15 9 2 3 4 (1) 15 3 (0) 3 =( α− α )b5/2 + (− − α + α )b5/2 64 64 32 64 32 9 3 15 3 (0) 3 15 (1) α )b5/2 + (− − α2 − α4 )b5/2 =( α+ 8 8 4 8 4 9 2 (1) = α b5/2 16 3 9 3 (1) 15 (0) α )b5/2 = − α2 b5/2 + ( α + 32 16 16 45 2 (0) 9 9 3 (1) = α b5/2 + (− α − α )b5/2 128 64 64 3 3 3 (0) (1) = α2 b5/2 + (− α − α3 )b5/2 8 4 4 3 3 15 (0) (1) = − α2 b5/2 + ( α + α3 )b5/2 4 4 4 9 2 (0) 3 3 3 (1) = α b5/2 + (− α + α )b5/2 128 64 64 3 3 21 2 (0) (1) α b5/2 + (− α − α3 )b5/2 = 8 4 4 3 3 3 (0) (1) = α2 b5/2 + ( α + α3 )b5/2 8 8 8 3 3 3 (1) 9 2 (0) α b5/2 + ( α − α )b5/2 = 128 64 64 15 3 3 (1) 9 (0) = − α2 b5/2 + ( α + α )b5/2 32 16 16 9 (0) = α2 b5/2 8 = (−

We then give the expression of the partially reduced Hamiltonian, obtained using the equations (20,21).

H2, +

PR

= K( ¯2 + X ¯ 1 X2 ) C2 (α) (X1 X 1 ¯1 + X ¯ 2 X2 ) − 2(2 + k −1 + k)Y Y¯ ) ) C3 (α) ( (X1 X 2

articleMRL2.tex; 15/11/2001; 17:51; p.36

37

H4, + + + +

= C4 (α) C5 (α) C6 (α) C7 (α) C8 (α)

+ + + + + + + + + +

C9 (α) C10 (α) C11 (α) C12 (α) C13 (α) C14 (α) C15 (α) C16 (α) C17 (α) C18 (α)

PR

K( ¯ 1 Y Y¯ − X1 X ¯ 2 Y Y¯ ) (− X2 X 2 ¯ ¯ ¯ 2) ( X2 X1 X2 + X1 X2 X 2 2 2 ¯ ¯ ¯2) ( X1 X2 X1 + X1 X1 X ¯ 1 Y Y¯ + X1 X ¯ 2 Y Y¯ ) (k −1 + k)(X2 X −1 2 ¯1 X ¯ 2 Y + X1 X2 Y¯ 2 ) ((2 + k + k )(X ¯ 2 Y Y¯ + 2 X2 X ¯ 1 Y Y¯ ) +2 X1 X −1 2 ¯ (−1)(2 + k + k)(X2 Y 2 + X22 Y¯ 2 ) ¯ 2 + X2 X ¯ 2) ( X22 X 1 1 2 −1 ¯ 1 Y Y¯ + k X2 X ¯ 2 Y Y¯ ) ( k X1 X (−2)(k + k −1 )Y 2 Y¯ 2 ¯ 2) ( X12 X 1 2 (2 + k + k −2 )Y 2 Y¯ 2 ¯ 2 Y Y¯ − 2 X1 X ¯ 1 Y Y¯ ) (−2 X2 X 2 2 ¯ ) ( X2 X 2 ¯ 2 Y 2 + X 2 Y¯ 2 ) (−1)(2 + k −1 + k)(X 1 1 −1 ¯ ¯ (− k X2 X2 Y Y + 4 Y 2 Y¯ 2 ¯ 1 Y Y¯ ) ¯ 2 − k X1 X ¯1 X + 41 X1 X2 X

We can remark that H4, P R does not depend on the parameter C3 (α) as we could have expected, coming from the term H2, N R . Indeed, the term in C3 (α) is equal to zero for the degree 4 but such terms appear for the degree 6 and higher. We finally give the analytical expression of the totally reduced Hamiltonian, taking into account that the parameter D is of the same order that the variables X. This expression is obtained using the change of variables (27). We can compare this expression with the expression given in (Robutel, 1995). In this paper, the expansion of the series is made with the use of the parameter D2 that has the same order in eccentricities and inclination as D2 . We can thus compare the term of order zero in D2 and D. In (Robutel, 1995), there was some misprints (0) (0) (0) for the terms H1,1,0,0 , H1,2,1,0 and H0,2,2,0 (see erratum in this same issue). The variables Xj of our study are not exactly the same than in this paper (see section 3.2, equations 29 and 30) and it explain why there are some change of sign in the expressions of the coefficients (r) Hp,¯p,q,¯q .

articleMRL2.tex; 15/11/2001; 17:51; p.37

38 The part of the totally reduced secular Hamiltonian is written under the form : Z X a2 (r) −2 ¯ p¯X q X ¯ q¯ (2π) Hp,¯p,q,¯q (α, k)Dr X1p X dλ1 dλ2 = 1 2 2. ∆ p,¯ p,q,¯ q ,r As the Hamiltonian is a function taking only real values, we have the relation among the terms (r)

(r)

Hp,¯p,q,¯q (α, k) = Hp¯,p,¯q,q (α, k) and only half of them are reported here. Coefficients of the quadratic part, H2,

TR

1 (1) (0) (0) H1,1,0,0 (α, k) = H0,0,1,1, (α, k −1 ) = (2 + k)αb3/2 (α) 8 3 (0) 1 (0) (0) (1) H1,0,0,1 (α, k) = H0,1,1,0 (α, k) = − αb3/2 (α) + (1 + α2 )b3/2 (α) 8 4 1 (2) (1) H0,0,0,0 (α, k) = − (2 + k + k −1 )αb3/2 (α) 2 Coefficients of the part of degree four, H4, (0)

(0)

(0)

(0)

TR

H3,1,0,0 (α, k) = H2,0,1,1 (α, k −1 ) 15 (0) (1 + k)α2 b5/2 (α) =− 128 3 (1) + (1 + k)(3α + α3 )b5/2 (α) 64 H1,1,0,2 (α, k) = H0,0,1,3 (α, k −1 ) 15 (0) (1 + k)α2 b5/2 (α) =− 128 3 (1) + (1 + k)(α + 3α3 )b5/2 (α) 64 9 (0) (0) (1) H2,1,1,0 (α, k) = H0,1,1,2 (α, k −1 ) = − (1 + k)α2 b5/2 (α) 64 15 15 (0) (0) H1,2,1,0 (α, k) = (− (α + 3α3 ) − k(α + α3 ))b5/2 (α) 64 32 3 3 (1) +( (2 + 3α2 + 6α4 ) + k(2 + 3α2 + 2α4 ))b5/2 (α) 64 32

articleMRL2.tex; 15/11/2001; 17:51; p.38

39

(0)

15 15 (0) (3α + α3 ) − k −1 (α + α3 ))b5/2 (α) 64 32 3 3 (1) +( (6 + 3α2 + 2α4 ) + k −1 (2 + 3α2 + 2α4 ))b5/2 (α) 64 32 3 (0) = (5k −1 + 24 + 5k)α2 b5/2 (α) 64 3 (1) + (k −1 + k)(α + α3 )b5/2 (α) 32 3 (0) (6 + 18k + 7k 2 )α2 b5/2 (α) = 128 3 (1) + (2α + 4α3 − (α + α3 )k 2 )b5/2 (α) 64 9 45 2 (0) (1) α b5/2 (α) − (α + α3 )b5/2 (α) = 128 64 3 (0) (7k −2 + 18k −1 + 6)α2 b5/2 (α) = 128 3 (1) + (4α + 2α3 − (α + α3 )k −2 )b5/2 (α) 64 15 3 9 (0) (1) = (2 + k + k −1 )( α2 b5/2 − ( α + α3 )b5/2 ) 32 16 16

H0,1,2,1 (α, k) = (−

(0)

H1,1,1,1 (α, k)

(0)

H2,2,0,0 (α, k)

(0)

H0,2,2,0 (α, k) (0)

H0,0,2,2 (α, k)

(2)

H0,0,0,2 (α, k) (2)

(2)

H0,0,1,1 (α, k) = H1,1,0,0 (α, k −1 ) = 3 (0) − (7k −2 + 19k −1 + 17 + 5k)α2 b5/2 16 3 −2 (1) + (k + k −1 − 1 − k)(α + α3 )b5/2 ) 8 9 (1) (2) H0,1,0,1 (α, k) = (2 + k + k −1 )α2 b5/2 16 15 (2) (0) H0,1,1,0 (α, k) = (2 + k + k −1 )( (α + α3 )b5/2 8 3 9 3 (1) +(− − α2 − α4 )b5/2 ) 4 8 4 3 9 15 (0) (1) (2) H0,2,0,0 (α, k) = (2 + k + k −1 )( α2 b5/2 − ( α + α3 )b5/2 ) 32 16 16 3 (4) (0) H0,0,0,0 (α, k) = (7k −2 + 20k −1 + 26 + 20k + 7k 2 )α2 b5/2 8 3 −2 (1) − (k + 2k −1 + 2 + 2k + k 2 )(α + α3 )b5/2 4

articleMRL2.tex; 15/11/2001; 17:51; p.39

40 Acknowledgements We would like to thank Mickael Gastineau for his help in developping the computer algebra system TRIP which enables us to perform practically this reductions. And, as for the theoretical part, we would like to thank Khaled Abdullah, Alain Albouy and Alain Chenciner for very useful discussions about the partial reduction.

References Abdullah, K. and Albouy, A. : 2001, ’On a strange resonance noticed by M. Herman’, preprint Arnold, V.I. : 1963, ’Small denominators’, Russ. Math. Survey 18 pp. 86-192. Arnold, V.I., Kozlov, V.V. and Neishtadt A.I. : 1985, ’Mathematical Aspects of Classical and Celestial Mechanics’, Springer second edition, 1997, pp. 78-106. Bennett, T.L. :1905, ’On the reduction of the problem of n bodies’, Messenger of mathematics Vol.XXXIV pp. 113-120. Birkhoff, G.D. : 1927, ’Dynamical Systems’, A.M.S. Coll A.M.S. Coll. Publications 9 (Rhode Island, Providence: American Mathematical Society) Boigey, F. : 1981, ’Transformations canoniques ` a variables impos´ees. Applications a la m´ecanique celeste’, Th`ese, Universit´e Pierre et Marie Curie, Paris 6 pp. ` 113-114. Brumberg, V.A. : 1980, ’Analytical algorithms in celestial mechanics’, Naouka, Moscou (in Russian) Deprit, A. : 1969, ’Canonical transformation depending on a small parameter’, Celest. Mech. Dynam. Astron. Vol.1 pp. 12-30. Deprit, A. : 1983, ’Elimination of the nodes in problems of N bodies’, Celest. Mech. Dynam. Astron. Vol.30 pp. 181-195. Hagihara y. : 1989, ’Celestial Mechanics’, The MIT Press Vol.1 pp. 476-482. Jacobi, C.G.J. : 1842, ’Sur l’´elimination des noeuds dans le probl`eme des trois corps’, Astronomische Nachrichten Bd XX pp. 81-102. Lagrange, J.L. : 1772, ’Essai sur le probl`eme des trois corps’, Oeuvre compl`etesVol.6 pp. 229-324. Laskar, J. : 1984, ’Th´eorie g´en´erale plan´etaire : ´el´ements orbitaux des plan`etes sur un millions d’ann´ees’, Th`ese, Observatoire de Paris. Laskar, J. : 1985, ’Accurate methods in general planetary theory’, Astronomy and Astrophysics 144 pp. 133-146. Laskar, J. : 1987, ’Secular evolution of the solar system over 10 million years’, Astronomy and Astrophysics 198 pp. 341-362. Laskar, J. : 1989, ’Syst`emes de variables et ´el´ements’ in ’Les m´ethodes modernes de la m´ecanique’, D. Benest and C. Froeschl´e (Eds), Editions Fronti`eres pp. 63-87. Laskar, J. : 1989, ’Manipulation de s´eries’ in ’Les m´ethodes modernes de la m´ecanique c´eleste’, D. Benest and C. Froeschl´e (Eds), Editions Fronti`eres pp. 89-107. Laskar, J. and Robutel P. : 1995, ’Stability of the planetary three body problem. I. Expansion of the Planetary Hamiltonian’, Celest. Mech. Dynam. Astron. Vol.62 pp. 193-217.

articleMRL2.tex; 15/11/2001; 17:51; p.40

41 Laskar, J. : 2001, ’AMD-stability and the organization of planetary systems’, preprint. Malige, F. : 2001, ’Stabilit´e effective des syst`emes plan´etaires’, Th`ese, Observatoire de Paris. Meyer, K.R.: 1973, Symmetries and Integrals in Mechanics, Dynamical Systems Academic Press, New-york pp. 259-271. Meyer, K.R. and Hall G.R. : 1992, Introduction to Hamiltonian Dynamical Systems and the N-Body Problem, Springer-Verlag. Milani, A. and Knezevic, Z.: 1990, ’Secular Perturbation Theory and Computation of Asteroid Proper Elements ’, Celest. Mech. Dynam. Astron. Vol.49 pp. 347-411. Poincar´e H. : 1892, ’M´ethodes nouvelles de la m´ecanique c´eleste’, Gauthier-Villars Vol.7 pp. 7-47. Poincar´e H. : 1896, ’Oeuvres completes de Poincar´e’, Gauthier-Villars Vol.7 pp. 496-511. Robutel P. : 1993, ’Contribution ` a l’´etude de la stabilit´e du probl`eme des trois corps’, Th`ese, Observatoire de Paris pp. 8-9. Robutel P. : 1995, ’Stability of the planetary three-body problem. II. KAM theory and existence of Quasiperiodic Motions’, Celest. Mech. Dynam. Astron. Vol.62 pp. 193-217. Spivak M. : 1999, ’A comprehensive Introduction to differential geometry’, Publish or Perish INC, Houston, Texas Vol.1.

articleMRL2.tex; 15/11/2001; 17:51; p.41