A first integral to the partially averaged Newtonian potential of the three-body problem

We consider the partial average i.e., the Lagrange average with respect to {\it just one} of the two mean anomalies, of the Newtonian part of the perturbing function in the three--body problem Hamiltonian. We prove that such a partial average exhibits a non--trivial first integral. We show that this integral is fully responsible of certain cancellations in the averaged Newtonian potential, including a property noticed by Harrington in the 60s. We also highlight its joint r\^ole (together with certain symmetries) in the appearance of the so called"Herman resonance". Finally, we discuss an application and an open problem.


Motivation
The purpose of this work is to highlight a property of the "partial average of the Newtonian potential" and discuss some consequences.
There are many examples, in Celestial Mechanics, of canonical maps of the form above. Well known ones are the above mentioned Delaunay map (hereafter, D), or the coordinates after the Jacobi-Deprit reduction 3 of the nodes (J ) (Jacobi, 1842;Deprit, 1983). Another example, called "perihelia reduction" (P), has been introduced by . A comprehensive review can be found in Pinzari (2015). All the maps mentioned here might actually be named double Kepler maps, since, in such cases, they satisfy (1)-(2), with, in turn, the (u, v)'s having the form (u, v) = (Λ 1 , ℓ 1 ,û,v) , du ∧ dv = dΛ 1 ∧ dℓ 1 + dû ∧ dv where ℓ 1 ∈ T and Λ 1 is such that (2) holds also with m 2 , M 2 , y (2) , x (2) replaced by m 1 , M 1 , y (1) , x (1) . In Section 2 below we shall present a "genuine" partial Kepler map, namely a map C where (2) holds only for one of the bodies.
We have been interested to the function (3) because, in planetary (1+N )-body theories, one has to deal with analogue maps of the kind C N : (Λ, ℓ,û,v) ∈ A N × T N × W → (y (1) , · · · , y (N ) ), (x (1) , · · · , x (N ) ) with W a domain in R 4N in terms of which the Hamiltonian of the system is where the Hamiltonian is composed of a leading"Keplerian part", given by − N i=1 , slightly perturbed by a function f N . The splitting (4) is possible -and in fact it has been adopted by Arnold (1963); Laskar and Robutel (1995); Chierchia and Pinzari (2011); Palacián (Sayas and Yanguas); Meyer (Palacián and Yanguas); , for example, in the so-called planetary problem where one of the masses ("sun") is much larger than the remaining, equally sized, N ones ("planets"). In that case, averaging over the Keplerian frequency vector leads to study the so-called "secular problem" where the perturbing term is given by "multi-averaged Newtonian potential" (the study of which goes back to Sundman (1916)) . (6) It is known (Gallavotti, 1986) that the dynamics of the full problem is well approximated by the one of the secular one as soon as no resonances between the frequencies (5) appear. In case of resonance, for example, in the case N = 2, with the two planets being much distant one to the other, it is reasonable to expect that a better approximation is obtained replacing the average (6) with the partial average (3). Concretely, it might be challenging to investigate whether there is an application to any of the following regions of motion that have been proposed by Féjoz (2002, p. 310), for the N = 2 case: -the planetary region, where the eccentricity of the outer ellipse and both semi-major axes are in a small compact set, and two masses are small compared to the third mass; -the lunar region, where the masses are in a compact set, and the outer body is far away from the outer two; -the anti-planetary region, where the outer body ellipse may have a large mass, provided its ellipse is far away from the outer two; -the anti-lunar region, when the ellipses of the two outer bodies are close, but the corresponding masses are much different.
We now go back to h 2 in (3). We firstly observe that Theorem 1 h 2 is integrable by quadratures.
Indeed, h 2 has six degrees of freedom and possesses, besides itself, the following five commuting 4 integrals: I 1 := the semi-major axis action Λ 2 := m 2 √ M 2 a 2 ; I 2 := the Euclidean length x (1) of x (1) ; I 3 := the Euclidean length of the total angular momentum C := C (1) + C (2) , with C (i) := x (i) × y (i) , and "×" denoting skew-product; I 4 := its third component; I 5 := the projection of the angular momentum C (2) along the direction x (1) . Indeed, I 1 is trivially due to the ℓ 2 -averaging; I 3 and I 4 descend from the invariance by rotations of h 2 ; I 2 and I 5 from invariance by rotations around the x (1) axis. Such integrals are independent if C (1) and C (2) are not parallel. Otherwise, the problem reduces to be planar, namely, h 2 has four degrees of freedom, and three independent commuting integrals are obtained neglecting, in the list above, I 4 and I 5 .
Remark 1 The list of independent first integrals to h 2 is even longer then the one above. For example, in the spatial case, the three components of x (1) and the three components C (2) are all first integrals. However, the maximum number of commuting first integrals that can be formed with these quantities is four (and the functions I 2 , I 3 , I 4 and I 5 are an example of them).
Remark 2 The integrability of h 2 does not imply that also the partial average of the three-body problem Hamiltonian is so, because this includes also a kinetic term. This is an even different situation compared to the secular problem mentioned above, whose non-integrability is clearly proven, as a consequence of the so-called splitting of separatrices (Féjoz and Guardia, 2016)). since E 0,C is ℓ 2 -independent. This is the thesis.

⊓ ⊔
In the next sections we highlight some properties of the partially averaged Newtonian potential that descend from Theorem 1 and Theorem 2. More precisely, the paper is organised as follows. In Section 2 we show that, as a consequence of Theorem 2, an infinite number of Fourier coefficients in the expansion of h 2 with respect to the perihelion of its outer planet cancel. This property is a generalisation of a fact noticed by S. Harrington in the 1960s Harrington (1969). To this purpose, we introduce a set of canonical coordinates in terms of which h 2 and E 0 are reduced to one degree of freedom. In Section 3 we show that there is an explicit functional dependence between h 2 and E 0 . We call this circumstance "renormalizable integrability". The author argues that it might be helpful in the framework of the study of the three-body problem. For example, it would be nice to understand if fixed points of E 0 , both of elliptic and hyperbolic character, being at the same time fixed points to h 2 with the same character, might give rise to quasi-periodic motions in the three-body problem; if hyperbolic equilibria might lead to a splitting of separatrices, etc. . Instead of addressing such issues here (which would lead much further than the purposes of this note; see, however,  for an application in this direction), we discuss the relations between level curves and the fixed points of the two functions. Next, we show that, as a consequence of renormalizable integrability and the well known A linear combination with integer coefficients in a suitable expansion of h 2 is identically verified. We name it "generalised Herman resonance" since it recalls the well known Herman resonance in the doubly averaged Newtonian potential (we refer to Abdullah and Albouy (2001) or Féjoz (2004, Proprieté 80) for informations on Herman resonance). After proving, in Section 4, an algebraic property of the well known Legendre polynomials (which, roughly, says that a certain average of a Legendre polynomial is still a Legendre polynomial), we establish, in Section 5, a link between the aforementioned generalised Herman resonance and Herman resonance. In this conclusive section we also provide a sort of "eccentricity-inclination" expansion at any order for such function and discuss a problem which is left open.

Generalised Harrington property
In this section we assume that the map C in (1) includes, among the u's, the impulse u 1 := G 2 := C (2) . We also give x (1) , x (2) the meaning of "interior", "exterior" planet, respectively, because we write formal expansions with respect to x (1) . We prove the following Theorem 3 Fix a domain for C where x (1) C × P (2) C , and C (2) where ρ(Λ 2 , u, v) := x (1) C and ϕ(Λ 2 , u, v) is the angle formed by the two vectors x (1) (2) C , with respect to the counterclockwise orientation established by C (2) C . Assume also that h nm depends on (Λ 2 , u, v) only via the following quantities with h 0m being independent of u 1 = G 2 for all m ≥ 0. Then In the case that h nm = 0 for n − m odd, for n ≥ 1, the following stronger identities hold: To prove Theorem 3, we shall need the following and assume that ∂ Γ a ≡ 0 and h 0m is independent of Γ for all m ≥ 0. Then h nm = 0 for all m ≥ max{1, n}.
Proof Due to the assumptions of h and g, their Poisson parenthesis at left hand side of (13) is a Fourier series including only sines {sin kγ} k≥1 . Projecting (13) over such basis, we obtain the following relations: for all n = 0, 1, m = 1, 2, · · · , where δ ij is the Kronecker symbol, and h −1,k := 0 for all k ∈ Z. We now prove that such relations imply h nm ≡ 0 for m ≥ max{1, n}. We proceed by steps.
(iii) We prove h nm = 0 for n ≥ 1 and m ≥ n. We proceed by induction on n.
The case n = 1 has been done in (ii). We assume that it is true for n ≥ 1 and prove it for n + 1. We use (14) replacing n with n + 1 and taking m ≥ n + 1: Here we have used that for m ≥ n + 1, m + 1 > m − 1 ≥ n, so the first line and the two first terms in the second line vanish. The last term also vanishes because m ≥ n + 1 ≥ 2, so the Kronecker symbol is zero. The lemma is completely proved.

⊓ ⊔
We now proceed to prove Theorem 3. To this end, we introduce a specific system of canonical coordinates which will allow us to apply the lemma above.
The K-map Define the "nodes" ν 0 := k × C , ν 1 := C × x (1) , ν 2 := x (1) × C (2) , ν 3 := C (2) × P (2) and assume that they do not vanish. Denote, as above, as P (2) , with P (2) = 1 the direction of its perihelion (well defined because the eccentricity does not vanish), a 2 its semi-major axis, we define the map via the relations where (i, j, k) is a prefixed reference frame, and for u, v ∈ R 3 lying in the plane orthogonal to a vector w and α w (u, v) denotes the positively oriented angle (mod 2π) between u and v (orientation follows the "right hand rule"). We remark that the planar case correspond to taking Θ = 0 and ϑ = π (prograde case) of ϑ = 0 (retrograde case). The map K verifies (1)-(2) with ℓ 2 = l 2 and u = (G 2 ,ȗ), v = (g 2 ,v), wherȇ u = (Z, G, R 1 , Θ),v = (z, g, r 1 , ϑ). Therefore u and v are also as claimed in the assumptions of Theorem 3. The canonical character of the coordinates K is discussed in  and to such paper we refer also for the formula, in terms of K, of the function E 0 in (7), which is We continue denoting as h the function in the statement expressed in terms of K. It follows from the definitions (15) that ρ = r 1 and ϕ = g 2 , so, by (9), h is given by Here we have used that, by assumption, the coefficients h nm in this expansion depend only on Λ 2 , G 2 , Θ. Therefore, in terms of K, the assumption that h Poisson commutes with E 0 reduces to Furthermore, with a = G 2 2 , we have ∂ G2 a ≡ 0 and, finally, h 0m is independent of G 2 for all m ≥ 0, being this one of the assumptions of Theorem 3. We can thus apply Lemma 1 and we obtain that h nm (r 1 , Λ 2 , Θ) ≡ 0 for m ≥ max{1, n}. The identities (12) trivially follow, under the additional assumption that h nm = 0 if n − m is odd.
⊓ ⊔ Application of Theorem 3 to the function h 2 In this section we discuss the application of Theorem 3 to the function h 2 in (3). First of all, h 2 Poisson commutes with E 0 , as stated by Theorem 2. As in the proof of Theorem 3, we now write h 2 in terms of the coordinates K in (15) and we fix a domain as in the statement of the theorem. This map is useful because ρ = r 2 , ϕ = g 2 and the functions in (10) are coordinates in such system, so we have only to check that h 2 affords an expansion of the form: with ∂ G2 h 1,0m (Λ 2 , Θ, G 2 ) ≡ 0. We shall also check that, in this summand, only terms with even n − m appear. We observe that, since h commutes with I 1 , · · · , I 5 , and, by their definitions, such functions are coordinates in the system K: then we have that h 2 is independent of their conjugate coordinates, respectively, ℓ 2 , R 1 , g, z, ϑ. The angles g, z are themselves first integrals to h 2 and so we have that h 2 is also independent of G, Z. In summary, h 2 will be a function of r 1 , Λ 2 , Θ, G 2 , g 2 only. Now we check that h 2 affords an expansion of the form (9), with h nm depending only on the quantities (10). As already observed in the proof of Theorem 3, in terms of the coordinates K, this reduces to check that h 2 , in terms of K has an expansion of the form (17). To this end, we start from the expansion of the Newtonian potential in Legendre polynomials (see Section 4) In terms of K, such quantities are given by is the true anomaly. The two former expressions are classical; the one for t has been worked out by . Inserting these expressions into (19) and taking the l 2 -average 6 , we have that with This expression shows h 2,n (Λ 2 , Θ, G 2 , g 2 ) is even in g 2 : so, it affords a Fourier expansion and the claimed expansion (17), follows. We finally check that ∂ G2 h 1,0m ≡ 0 for all m ≥ 0. But this is a consequence of the fact that, for r , which is Γ -independent by Proposition 1. Then the assertion and hence the thesis (11) holds. We now check that, in the case of h 2 , one also has h 2,nm = 0 for n − m odd, so, for n ≥ 1, the stronger identity in (12) holds. Denoting as c np ∈ Q the coefficients in the expansion where, we recall, only p's having the same parity as n appear (an explicit formula for the c np 's is available from the first formula in Equation (53) below), so that 6 Recall the well known transition formula (see, e.g., Palacián (Vanegas and Yanguas)) Using the expansion cos p (g 2 + f 2 ) = (cos g 2 cos f 2 − sin g 2 sin f 2 ) p = p k=0 (−1) k p k sin k g 2 cos p−k g 2 sin k f 2 cos p−k f 2 and finally inserting this expression into (24) and afterwards into (21), we can write (21) as a trigonometric polynomial in g 2 having degree n given by The function under the integral in the expression above has the same parity as k, soĥ npk vanishes for k odd 7 . Therefore, in the summand in (25) only even indices k appear. But for any even k, sin k g 2 cos p−k g 2 has a Fourier expansion p m=0 b m cos mg 2 where m has the same parity as p, which is the same as n. We collect all of the information in the following Proposition 2 All the assumptions if Theorem 3 are verified with h = h 2 . Therefore, the coefficients h 2,nm in the expansion (9) verify (11) and, for n ≥ 1, they verify the stronger identity (12). Choosing C = K, the expansion in (20)-(23) holds true, with h 2,nm verifying (11) and (12). In particular, the term h 2,1 vanishes identically and h 2,2 , called the dipolar term, does not depend on g 2 .
One could ask what the last assertion becomes when using, instead of the Kmap, one of the more familiar maps, D, J or P, mentioned in the introduction. As a matter of fact, the same assertion holds, apart from parity in the Fourier expansion: 7 Incidentally, by explicit computation of the integral, we obtain, for even k, where only terms with j having the same parity as p appear.
Proposition 3 Let g D 2 , g J 2 or g P 2 denote the angles conjugate to G 2 , in the case of the maps D, J or P. In the expansion the coefficients h 2,n afford a Fourier expansion +∞ m=0 (a nm cos g C 2 +b nm sin g C 2 ), with m having the parity as n and a nm , b nm verifying (11) and (12). In particular, h 1,1 ≡ 0 and h 1,2 does not depend on g D 2 , g J 2 or g P 2 , respectively.
Proof The maps D, J or P share the property that u 1 = G 2 is one of their impulses. However, the coordinate conjugate to G 2 is different in any of such cases, and is given by the angle, that here we denote as g D 2 , g g 2 or g P 2 , formed by a certain "node" (we call so a non-vanishing vector in R 3 ) with P (2) in the plane orthogonal to C (2) , with respect to the positive direction determined by C (2) . The mentioned node is given by: where P (1) denotes the direction of the perihelion associated to the Keplerian ellipse of the inner body. We then find the following relation where ϕ C is the angle determined by ν C and ν 2 in (15). Such function does not depend on g C 2 . Since the functions a 2 , Θ, r 1 , ζ 2 , f 2 in (20), expressed in terms of D, J , P, even do not depend on g C 2 , the proof of Proposition 3 follows, replacing such functions into (20), and using the information given by Proposition 2.

Renormalizable integrability
Another consequence of Theorems 1 and 2 is that there actually exists a functional dependence between h 2 and E 0 which we shall write explicitly. To this end, we premise some abstract consideration.
Proposition 4 If h is renormalizably integrable via g, then: (i) I 1 , · · · , I n are first integrals to h and g; (ii) h and g Poisson commute.
Proof It follows from (26) that In this expression, all the terms in the summand vanish because of (28), while the last term vanishes because of (29): This proves (ii). (i) follows from (ii), replacing the couple (h, g) with (h, I i ) or (g, I i ), with i = 1, · · · , n.
⊓ ⊔ At level of motion, renormalizable integrability can be rephrased as follows.
Proposition 5 Let h be renormalizably integrable via g. Fix a value I 0 for the integrals I and look at the motion of (y, x) under h and g, on the manifold I = I 0 . For any fixed initial datum (y 0 , x 0 ), let g 0 := g(I 0 , y 0 , x 0 ). If ω(I 0 , g 0 ) := ∂ gh (I, g)| (I0,g0) = 0, the motion (y h (t), x h (t)) with initial datum(y 0 , x 0 ) under h is related to the corresponding motion (y g (t), x g (t)) under g via In particular, under this condition, all the fixed points of g in the plane (y, x) are fixed point to h. Values of (I 0 , g 0 ) for which ω(I 0 , g 0 ) = 0 provide, in the plane (y, x), curves of fixed points for h (which are not necessarily curves of fixed points to g).
⊓ ⊔ Below, we prove that, under an additional condition, the converse of Proposition 4 holds true.
Theorem 4 Let h, g two commuting functions of the form (26) on the possibly complex domain D as in (27), with I i pairwise Poisson commuting. For any fixed c = (c 1 , · · · , c n ) ∈ I(B), let ∆ c be the set of stationary points of the function (y, x) → g(y, x, c 1 , · · · , c n ), and put U * c := U \ ∆ c Assume that the set D * := (p,q)∈B (p, q) × U * I(p,q) has full closure. Then h is renormalizably integrable via g.
We have that h is x-independent, while h ′ is y-independent. Let us check the assertion for h (for h ′ is specular). Again by the Implicit Function Theorem: because of (31). Choosing, for a fixed (p, q) ∈ B, (y, x) ∈ U * I(p,q) , c 1 = I 1 (p, q), · · · , c n = I n (p, q), c n+1 = g(p, q, y, x), we have the thesis on the set D * . Then, by smooth continuation, the thesis holds on all of D = B × U .

⊓ ⊔
Remark 3 We observe that the proof is constructive: it provides the function h via the formulae (32)-(33).
In the following, we prove that h 2 is renormalizably integrable via E 0 as an application of Theorem 4. Afterwards, in Section 3.1, we shall exhibit, explicitly, the relative function h 2 realizing (29). In Section 3.2, as a counter-example to the last assertion of Proposition 5, we shall exhibit a curve of fixed points for h 2 which is not so for E 0 .
Application of Theorem 4 to h 2 and E 0 . We aim to apply Theorem 4 to h 2 and E 0 . As in the former section, we use the coordinates K defined in (15). This map turns to be useful, because the integrals I 1 , · · · , I 5 are coordinates of such system and hence depend on (p, q) only via one of the p's or one of the q's: see (18). As a first step, we aim to check that h 2 and E 0 have the form in (26), with n = 3 , I = (I 1 , I 2 , I 3 ) = (r 1 , Λ 2 , Θ) , y = G 2 , x = g 2 The expression of E 0 has been given in (16), so it turns to be as claimed. The expression of h 2 in terms of K has been discussed by , and is where with ζ 2 , as above, the eccentric anomaly, and f 2 the true anomaly, both depending on (Λ 2 , G 2 , l 2 ). We observe that it is possible to have a closed formula for h 2 , since the integration in dl 2 can be written explicitly by means of the eccentric anomaly dl 2 = ̺ 2 dζ 2 and the true anomaly f 2 can be eliminated via the well known relation Then we rewrite h 2 as which is precisely of the form (26). As a second step, we check that, for any fixed value of the integrals I in (34), the set of fixed points of E 0 as a function of (G 2 , g 2 ) is at most one-dimensional in the plane (g 2 , G 2 ). Indeed, equations define an algebraic set in having positive co-dimension. Then Theorem 4 applies and we have the following Proposition 6 h 2 is renormalizably integrable via E 0 . Namely, there exists a function h 2 such that h 2 (r 1 , Λ 2 , Θ, G 2 , g 2 ) = h 2 r 1 , Λ 2 , Θ, E 0 (r 1 , Λ 2 , Θ, G 2 , g 2 ) .
In the two following sections we discuss some insights of dynamical character, related to the renormalizable integrability of h 2 .

The explicit expression of h 2
The function h 2 in Proposition 6 can be written explicitly, and this is the purpose of this section. Before doing it, let us premise some algebraic consideration.
Definition 2 (The class H * ) We call class H * the set of functions of the form f (a, b, u, v) = 1 2π T P (uc(w))dw a 2 + 2abQ(vs(w)) + b 2 P (uc(w)) 2 where: u → P (u), u → Q(u) are smooth functions for u = 0; P (0) > 0; u → Q(u) is odd; c, s are periodic functions such that there exist two "symmetries", i.e., transformations σ, σ ′ : T → T verifying |∂ w σ| = |∂ w σ ′ | ≡ 1 and Definition 2 implies that any f ∈ H * is homogeneous of degree −1 in (a, b); even in all of their arguments and, moreover, verifies Proposition 7 All the functions in H * afford a formal series expansion where q(a, b) is a positive definite quadratic form and p ij (a, b) are polynomials of degree 4(i + j − 1) with coefficients in Q, even separately in a and b. In particular, for any f ∈ H * , there exist r, s ∈ Q such that Remark 4 We call the identity (43) generalised Herman resonance and underline that its validity is strongly based on the identity (40). For the averaged Newtonian potential, (40) is guaranteed by the Keplerian property (Proposition 1).
Proof Using the formula (38), it is easy to prove, by induction, that any f ∈ H * affords an expansion of the kind where p ij (a, b) are polynomials in (a, b) and Using the parity of f with respect to all of its arguments, one has, actually, that p ij 's are even with respect to a and b separately, and vanish if i, j are not both even, so we have an expansion of the form (41), with f hk = f 2h,2k . Furthermore, since f is homogeneous of degree −1, all of its derivatives with respect to u or v are homogeneous of the same degree. Since q(a, b) is homogeneous of degree 2 (see (45)), we have that the p 2h,2k in (44) are to be homogeneous of degree 4(h + k). Finally, due to (40), p 2h,2k (1, 0) = p 2h,2kj (0, 1) ≡ 0 for all (h, k) = (0, 0). Combining this with parity of p 2h,2k with respect to a and b separately, (42), p 2h,2k (a, b) = a 2 b 2 p hk (a, b) where p(h, k)(a, b) has degree 4(h + k − 1). This proves the former assertion. The latter follows from this, since, when h + k = 1, with p 10 and p 01 having degree 0, namely, p 10 and p 01 ∈ Q. So, one can take r = −p 01 , s = p 10 . ⊓ ⊔ Let us now proceed to write down an explicit expression of function h 2 in Proposition 6. We let Note that U is in the class H * , with We prove that Proof Reasoning as in the proof of Theorem 4 (see Remark 3), we invert equation E 0 (r 1 , Λ 2 , Θ, G 2 , g 2 ) = E 0 with respect to G 2 in the complex field, fixing a value of g 2 . We choose g 2 = π 2 , so that cos g 2 = 0 and the the inversion is immediate: Then h 2 (r 1 , Λ 2 , Θ, E 0 ) is given by Using the formula in (36), we obtain with E, I as in (47).

⊓ ⊔
Remark 5 Combining Propositions 6, 8 with (7) and the definitions of G 2 and Θ in (15), we obtain that, for a generic C as in (1), In the Section 5.1, we shall use the following consequence of this.
Proposition 9 Let where U is a domain of R 4 , verify (2) and letṽ ∈ R 3 . Then E 2 := e 2 2,C2 + e 2,C2 v · P (2) C2 a 2 , where the sub-fix C 2 denotes the composition with C 2 . and (y (2) , x (2) )•C = (y (2) , x (2) )• C 2 , depending only on (Λ 2 , ℓ 2 , u, v). ⊓ ⊔ 3.2 A curve of fixed points for h 2 (which is not so for E 0 ) Proposition 6 implies that any level set (in the plane (G, g)) to E 0 is also a level set of h 2 and hence, in particular, any fixed point to E 0 is so to h 2 . Here we prove that the converse is not true: Proposition 10 If Θ = 0 and r 1 /a 2 is sufficiently small, in the plane (G 2 , g 2 ), there exists at least a curve of fixed points of h 2 which is a level set of E 0 , but is not a curve a fixed points to it.
Proof In principle, to find any such curve, one should solve equation ω(I,h) := ∂ gh (I,h) = 0. In the case of h 2 , such equation seems too difficult, so we shall use a perturbative approach. We look at the Taylor expansion (20) of h 2 in (35) in powers of r 1 . Letting ε := r1 a2 , we obtain By (48), a corresponding expansion for the function h in (49) is obtained letting G 2 = √ E 0 and g 2 = π 2 . We obtain: We study Equatioñ via the Implicit Function Theorem, for small ε. Neglecting the O(ε 2 ) inside parentheses, we obtain the solution The non-degeneracy condition at this solution is verified, since indeed Then for sufficiently small ε, Equation (51) has, as a solution, the following level set of E 0 : Replacing the formula for E 0 in (16), we rewrite such solution as the curve, in the (G 2 , g 2 ) plane, By Proposition 5, S is a curve of fixed points for h 2 . It remains to prove that S is not a curve of fixed points for E 0 . The fixed points of E 0 are the solutions of the system (37). The curve S includes a point having coordinates which does not solve the system (37) (it does not solve the second equation). ⊓ ⊔

Remark 6
The proof fails for Θ = 0, because, in such a case, the leading part in equation (51) has no solution.

An algebraic property of Legendre polynomials
The Legendre polynomials P n (t), with P 0 (t) = 1, P 1 (t) = t, · · · , are defined via the ε-expansion Many notices on such classical polynomials may be found in Giorgilli (2008, Appendix B).
The purpose of this section is to present an algebraic property of the P n 's. Roughly, it says that a certain average of a Legendre polynomial is still a Legendre polynomial. The author is not aware if it was known before and if there is a "dynamical" explanation of it.
Lemma 2 Let t ∈ R, |t| < 1, P n the n th Legendre polynomial. Then, where We shall prove Lemma 2 via the following one.
Lemma 3 The even Legendre polynomials P 2m (t) verify, for any h = 0, · · · , m, where τ := t 2 . In particular, the following relation holds Proof We first prove the former formula in (53). Let n ∈ N, k = 0, · · · , n with n − k even. We have Therefore, denoting as Π n the projection over the monomial ε n , Then the desired formula follows, taking n = 2m, k = 2h and noticing that The proof of the latter formula in (53) is a bit more complicate. We propose an algebraic one.
First of all, we change variable we are definitely reduced to prove the following identity To this end, we let g(ε, z) := 1 so that (analogously to (54)) we may identify We introduce the auxiliary functions so that g 1,0 = g. Observe that the linear space generated by such functions is closed under the derivative operation, since in fact D z g a,b (ε, z) = −εαg α+2,β+1 (ε, z) + βg a,b+2 (ε, z) .
More in general, by iteration, one finds where, from the identity one easily sees that the coefficients c corresponding to the case α = 1 , β = 0 .
⊓ ⊔ Proof of Lemma 2 Let Q n (t) denote the left hand side of (52). Observe that, since any P n (t) has the same parity, in t, as n and odd powers of cos θ have vanishing average, the Q 2m+1 (t)'s vanish, while the Q 2m (t)'s are polynomials of degree m in τ := t 2 . Since also the even Legendre polynomials P 2m 's are polynomial of degree m in τ , we only need to show, e.g., that The definition of Q 2m implies that, for h = 1, · · · , m Using Lemma 3, we find and hence the thesis follows. ⊓ ⊔

Applications
5.1 An explicit formula for a semiaxes-eccentricities-inclination expansion of a "mixed" averaged Newtonian potential.
Our purpose is to provide, under the previous assumptions, a representation formula for the function 8 The reader should not confuse the function h 12 above with what is commonly called "doubly averaged Newtonian potential", defined as (2) C even though, in the case that E 1 has identically vanishing eccentricity and C is regular in this limit, h 12 and h 12 coincide.
which we believe may turn to be useful in applications. We introduce the following Definition 3 For a given power series in the parameter ε g ε := ∞ n=0 a n ε n we denote as Π ε g ε the even power series with (−1)!! := 1.
We shall prove the following formula. We let U as in (46) and E(r 1 ) = e 2 2,C + r 1 e 2,C
The second-order term of this expansion of course exhibits (43), as a byproduct of Proposition 7 (because Π r kills the linear terms in r 1 in (66) acts on the even terms only modifying the coefficients). This identity reduces to the classical Herman resonance switching to Poincaré coordinates with the inner body moving on a circle. In this framework, Herman resonance naturally appears as a byproduct of parities (39), renormalizable integrability of the Newtonian potential (Proposition 8), Keplerian property (Proposition 1) and Lemma 2.
To prove Proposition 11, we need an equivalent formulation of Lemma 2, which is as follows.
In this section, we outline an underlying open problem in the framework of the paper. We pose a conjecture, that we aim to study in further work, which, if proved, may be applied to the two-centre Hamiltonian (8), so as to obtain a stronger assertion than Proposition 6. Throughout the section, V ⊂ R, U ⊂ R 2 are domains, (I, ϕ) ∈ I ×T, (p, q) ∈ U are pairwise conjugate canonical coordinates. We shall be concerned with realanalytic 9 functions ("Hamiltonians") for (I, ϕ, p, q, µ) ∈ P = I × T × U × (−µ 0 , µ 0 ) having the form: h k (I, p, q)µ k + O(µ p+1 ) ∀(I, ϕ, p, q) ∈ P .
The following result is well known and hence will be not discussed.
Lemma 4 Let h be in p-normal form and let g be a first integral of h. Then (i) g is in p-normal form; (ii) {h 1 , g} = O(µ p ), where h 1 (I, p, q, µ) := p k=1 h k (I, p, q)µ k 9 Following the standard terminology, a real function h is said to be real-analytic on a domain P ⊂ R p if there exists an open setP, with P ⊂P ⊂ C p , such that h has a holomorphic extension onP.
Corollary 1 For any p, g (p) tr (I, p, q) = g 0 (I, p, q)+ 1,tr (I, p, q)} = O(µ p ) We recall that h 1 (I, p, q, 0) = 1 2π 2π 0 h 1 (I, ϕ, p, q, 0)dϕ Definition 5 We shall refer to the formal series ∞ k=0 h k (I, p, q)µ k as perturbative series in µ to h. (70) has an independent first integral, its perturbative series converges, as well as the perturbative series to g. If h, g denote the sum of the two series, h is renormalizably integrable via g.

Conflict of interests
The author declares that she has no conflict of interest.