Birkhoff normalization, bifurcations of Hamiltonian systems and the Deprits formula

We consider Hamiltonian autonomous systems with n degrees of freedom near a singular point. In the case of absence of resonances of order less than or equal to 4 we present a direct computation of the Birkhoff normal form. In the case of two degrees of freedom, we study 1–parameter deformations of the 0 : 1, 1 : 1 and 2 : 1 resonant singularities. The obtained results are used in a direct derivation of the Deprits formula for the isoenergetic degeneracy determinant in the restricted three–body problem.


Introduction
Probably the most spectacular application of the Kolmogorov-Arnold-Moser (KAM) theory is in the restricted three-body problem (see [AKN, Mos, SiMo, Mar]). There one has a completely integrable Hamiltonian system with two degrees of freedom, which corresponds to fourth-degree Birkhoff normal form of the total energy near a Lagrangian libration point, and a perturbation, which corresponds to higher order terms. The invariant KAM tori, which fill a set of positive measure, guarantee the Lyapunov stability of this libration point. However, in order to use the KAM theorem, one has to check whether 588 W. Barwicz, M. Wiliński and H. Ż o1aødek JFPTA Birkhoff normal form and application of it to the restricted three-body problem. Recall also that the problem contains one parameter ζ defined by the ratio of masses of the heavy bodies. The formula for the coefficients in the Birkhoff normal form was obtained in explicit form by Leontovich [Leo], but in the three-body case he was able only to show that the quantity det is not identically zero as a function of the parameter ζ. The complete formula for det(ζ) was given by Deprit and Deprit-Bartholomé in [DDB] (see equation (4.9) below) and is cited in all sources about the subject. In practice, only few three-body systems are considered: Sun-Jupiter-Asteroid or Earth-Moon-Asteroid. However, more such situations in the celestial mechanics exist (with different values of ζ). Therefore, the formula for det(ζ) is potentially useful.
It is rather hard to repeat the Deprits' derivation of their formula, because the paper [DDB] does not contain clear ideas about it. We tried to reprove the result of [DDB], but in [BaZo] we succeeded only to obtain the Leontovich formula and to apply it for some special values of the parameter ζ. Our calculations disagreed with the Deprits formula (see Remark 3 below). But later, at a conference in Siedlce (Poland), we have learned from Markeev and Prokopenya that they have checked the calculations (also using some computer programs) and obtained the same result as in [DDB].
Therefore, the principal reason for writing this paper was to fix our mistakes and eventually to find a direct derivation of the Deprits formula (with an explanation of its amazing simplicity). The idea was to show that det(ζ), which is an algebraic function of ζ, is in fact a rational function of ζ 2 with possible poles corresponding to resonances of order 1, 2 and 3 and to ζ 2 = ∞. We prove it in Section 4.2. The next step is to analyze the situation near the distinguished values of ζ 2 .
In the case ζ 2 ≈ ∞, the eigenvalues of the corresponding linearization of the Hamiltonian system are not all imaginary, so a corresponding version of the Birkhoff normal form is needed. We do it in the next section. We do it in full generality, i.e., in the complex situation and with many degrees of freedom. Moreover, the derivation of the corresponding generalization of the Leontovich formula does not use iteration of the Birkhoff transformation. It is direct in the sense that it is reduced to a series of elementary substitutions. As a corollary, in Section 4.3 we find that the limit of det(ζ) as ζ → ∞ is finite.
The case of 0 : 1 resonance, ζ 2 = ζ 2 0 , corresponds to the situation when a pair of eigenvalues is zero and a 2-dimensional Jordan cell arises. We obtain an analogue of the Birkhoff normal form for the vector field and for its 1-parameter perturbation corresponding to variation of one of the small eigenvalues. A general theory is described in Section 3.1, where we show that the order of the pole of det(ζ) at ζ 2 0 is at most 2. In Section 4.4 we prove that in the three-body case det(ζ) is regular here (no pole).

Birkhoff normalization 589
case is simple and is shortly discussed in Section 3.3. In the case of 1 : 1 resonance we have two 2-dimensional Jordan cells and the analysis is slightly more involved, but here we need only to show that the corresponding pole is simple. We do it in Section 3.2. Section 4 is devoted to the restricted three-body problem. The function det(ζ) is a ratio P (ζ 2 )/Q(ζ 2 ) of two quadratic polynomials, where Q = (ζ 2 − ζ 2 1 )(ζ 2 − ζ 2 2 ) defines the resonances of order 2 and 3. We easily find (in Section 4.6) the value of the residuum of det(ζ) at ζ 2 2 . By a direct calculation (in Section 4.7) of det(ζ) at two special values of the parameter we find the three coefficients of P . In fact, we calculate det(ζ) for three values of ζ.

Birkhoff normal form
Consider a linear autonomous Hamiltonian system in C 2n , with respect to some linear symplectic structure on C 2n defined by a Poisson structure {·, ·}. Assuming that the matrix A = (a jk ) has pairwise different and nonzero eigenvalues, which appear in pairs λ j , −λ j , 1 we can diagonalize this system: here the functions (variables) z = {z j ,z j : j = 1, . . . , n} are related with the variables x = (x 1 , . . . , x 2n ) by means of a matrix B, i.e., x = Bz, such that the columns of B are the eigenvectors of A. The variables z satisfy the relations {z j , z k } = {z j ,z k } = {z j ,z k } = 0 for j = k, but the constants {z j ,z j } are not determined because the very variables z j andz j are defined up to multiplicative constants. Therefore, the quadratic Hamiltonian G (2) from (2.1) can be written in the form Generally, with the Hamilton function (2.3) we can associate the generalized actions J k ∈ C and generalized angles φ k ∈ C as follows: Note that each function J k is of rather special type: its zero locus consists of two transversal hyperplanes, but it has degenerate singularity at the origin (with infinite Milnor number). JFPTA Assume now that we have a holomorphic Hamiltonian system with n degrees of freedom and with a singular point at the origin of the type considered above. Therefore, we have We will use also the following notations: Additionally we assume the following: k j λ j = 0 for k j ∈ Z, |k j | = 1, 2, 3, 4, (2.6) i.e., the absence of resonances of order 1, 2, 3 and 4. (Here the case |k j | = 1 would correspond to a situation with a pair of zero eigenvalues and the case |k j | = 2 would correspond to a situation with two equal eigenvalues.) Example 1. If G is real and the eigenvalues are imaginary, λ k = − √ −1ω k = −iω k , ω k > 0, (i.e., when the origin is the so-called elliptic singular point), then there are natural symplectic variables In this case there exist so-called action-angle variables I k = (p 2 k + q 2 k )/2 = |z k | 2 /2 and ϕ k = arg z k . Then J k = iI k and H (2) = ±ω 1 I 1 ±· · ·±ω n I n . Note that using representation (2.3) we avoid the problem of signs before ω j I j .
If G is real and some eigenvalue λ k is real, then one can choose Here the analogue of the corresponding action-angle variables Vol. 13 (2013)

Birkhoff normalization 591
If n = 2, G is real and we have nonreal and nonimaginary eigenvalues λ = λ 1 = ν + iω, −λ, λ 2 =λ = ν − iω and −λ, then we can choose Then J 1 = J = 1 2 z 1z1 and J 2 =J 1 are nonreal and H (2) = 2 Re(λJ). (Here the notation Re has the standard meaning, the real part, in contrary to the notation RE.) The fourth order Birkhoff normal form [Bir] in the above situation is a symplectic change of variables such that The symplecticity of this change of coordinates implies preservation of the Poisson brackets: The main result of this section is the following theorem which generalizes a result by Leontovich [Leo].
Theorem 1. The coefficients in equation (2.8) are the following: Here h j ρσ (resp., h jk ρσ;ςτ or h jkl ρσ;ςτ ;υφ ) denotes h m with (m j ,m j ) = (ρ, σ) (resp., with given distinguished indices) and with zero other indices. JFPTA The proof we give below is essentially new and does not use generating function for the symplectic change; the standard (rather involved) proof for n = 2 can be found in [BaZo].

Lemma 1. The Birkhoff transformation is of the following form:
the a k j ,ã k j are constants (calculated below) and the dots denote inessential terms (of degree greater than or equal to 3).
The above form for the quadratic terms follows from the Poincaré-Dulac theorem and is unique (see [Arn1]). The distinguished cubic terms, like a j k z j z k 2 , are resonant in the Poincaré-Dulac sense. Therefore, they are not determined here. 2

Lemma 2. The action parts of the action-angle variables
(plus inessential quartic and higher order terms) where The Poincaré-Dulac theorem says that a systemẋ 1 = λ 1 x 1 + a 1 k x k , . . . ,ẋn = λnxn + a n k x k can be reduced to the so-called Poincaré-Dulac normal formẏ 1 = λ 1 y 1 + b 1 k y k , . . . ,ẏ n = λ n y n + b n k y k , where only resonant terms b j k y k (for which λ j = λ 1 k 1 + · · · + λnkn) in the jth equation remain.

Birkhoff normalization 593
(for k = l) and From this we arrive at the following lemma.
Lemma 3. We have (2.13) Indeed, from (2.11) we find because the cubic terms contribute completely to λ j J j . But this equals Let us pass to the determination of the coefficients a k j andã k j . Since the change from Lemma 1 must be symplectic, the sums a k j +ã k j , . . . follow from the following formula: (resp., α j;jkl ρσ;ςτ ;υφ ) denotes α j m with distinguished indices at the jth and kth places (resp., at the jth, kth and lth places).
Proof of Theorem 1. Using equations (2.10)-(2.13) we find contributions arising from different products h n h m . We begin with D 11 .
The changes (2.14) and (2.15) describe the change of the Birkhoff normal form as we move along a small loop around π −1 (Σ 0 ) (resp., around π −1 (Σ 1 )). Such a loop lies at a small disc intersecting transversally π −1 (Σ 0 ) JFPTA (resp., π −1 (Σ 1 )) at a point separated from other components of the bifurcational hypersurface π −1 (Σ) and from its singular locus (which correspond to multiple vanishing of eigenvalues and to multiple resonances). Therefore, they are the monodromy transformations corresponding to such loops.
Remark 2. The coefficients D ij are independent of the choice of the diagonalization coordinates, i.e., with respect to the action of the torus (C * ) 2n :

Bifurcations near resonant cases
In this section we consider Hamiltonian systems in C 4 (two degrees of freedom) near a singular point with a low-degree resonance and their 1-parameter deformations. We assume that the Hamilton functions depend analytically on the (complex) coordinates and on a complex parameter. These are deformations of Hamiltonian systems from the bifurcational hypersurfaces π −1 (Σ 0 ), π −1 (Σ 1 ) and π −1 (Σ 2 ) defined in Remark 1.

Jordan cell with zero eigenvalues
Here we consider the situation when n = 2 and λ 2 = 0 and λ 1 = 0, i.e., the 0 : 1 resonance. We have two possibilities: (i) the linear part, the matrix A, is diagonalizable, or (ii) the matrix A contains a 2-dimensional Jordan cell. Since the case (i) is rather straightforward and not needed for our aims, we assume the second possibility.
From [Arn2,Appendix 6] we learn that the quadratic part of the Hamilton function can be reduced to where μ = λ 1 , z = z 1 ,z =z 1 , κ = {z,z} (as before) and x, y are coordinates with {x, y} = 1. Indeed, such a Hamiltonian is a limit of a family of generic Hamiltonians such that the 2-dimensional subspaces corresponding to two pairs of eigenvalues ±λ 1 → ±μ and ±λ 2 → 0 are symplectic and skew orthogonal. Therefore, the (x, y)-plane which supports the Jordan cell is symplectic and skew orthogonal to the z-plane. (In the real case the sign before 1 2 x 2 is invariant with respect to real linear symplectic changes which preserve the form (3.1), but here we can put −1.) Assume the following expansion of the Hamiltonian:

Birkhoff normalization 597
Q =q 1 x +q 2 y, R = j+k=3 r jk z jzk and the dots denote inessential quartic and higher order terms. We apply the following symplectic change: where γ,γ, S,S are analogous like in Lemma 1 for n = 1. Note that the change (X, Y ) → (x, y) is a time 1 flow map g 1 F generated by the Hamiltonian (with the parameter Z 2 ) and the map (Z,Z) → (z,z) is an analogous map g 1 G generated by the Hamiltonian After this change we arrive at the following analogue of the Birkhoff normal form: and we adopt the same notations for · and RE as in the previous section.
Here the last term in B 5 comes from Theorem 1 for n = 1. Consider now a deformation H ε , ε ∈ (C, 0), of the Hamiltonian (3.2) such that the origin x = y = z =z = 0 is critical 4 and for ε = 0 the corresponding matrix A ε is nondegenerate. The natural deformation of the Jordan JFPTA cell is 0 ε −1 0 ; this is achieved by some genericity assumption ( d dε det A ε | ε=0 = 0) and eventual change of the parameter. In this case we can apply a symplectic change of the form (3.3) where the coefficients depend on ε and the terms of degree greater than or equal to 3 can be reduced to the form (3.4). Thus we arrive at the following proposition. Proposition 1. The family H ε can be reduced to the following normal form: where the coefficients μ = μ(ε), κ = κ(ε), B j = B j (ε) depend analytically on the parameter and B j (0) are given in equations (3.5).
Assume ε = 0. Let Thus we have The quadratic part of the Hamiltonian H Nor The cubic and quartic terms come from the substitution of (3.7) to H Nor ε . After applying Theorem 1 we arrive at the Birkhoff normal form (2.8) with (3.8) These formulas imply the monodromy transformation (2.14): the loop in the parameter space is {ε = ε 0 e iτ , τ ∈ [0, 2π]} for some small ε 0 > 0.

Pair of Jordan cells with nonzero eigenvalues
Here we assume λ 1 = λ 2 = μ = 0 and that the matrix A is not diagonalizable. (The case with diagonal A is the same as in Section 2.) From [Arn2] (see also [Mar, Dui, vdM]) we learn that, when the real matrix A has two imaginary eigenvalues ±iω of multiplicity 2 and A is not diagonalizable, then we can write H (2) = ± 1 2 q 2 1 + q 2 2 + ω(q 1 p 2 − q 2 p 1 ), or Vol. 13 (2013)

Birkhoff normalization 599
, v 2 } = 0 and {v 1 ,v 2 } = 2). (Like in the previous section, in the real case the sign before 1 2 v 1 2 is invariant under symplectic changes.) This suggests that we should take where the variables v j , v k obey the following Poisson brackets: (3.10) Then we get the system (3.11) i.e., with two Jordan cells.

600
W. Barwicz,M. Wiliński and H. Ż o1aødek JFPTA By construction this change is symplectic and the corresponding change in the Hamilton function is the following: plus nonlinear terms with respect to s m 's. We see that the corresponding (homological) linear operator H : These maps describe block operators between some subspaces of S 0 with fixed m 2 +m 2 (the sum of these indices decreases by 1). The distinguished operators are not surjective, because of the dimension counting (other block operators are surjective). It is easy to see that the subspaces complementary to images of the distinguished block operators are generated by V 2 4 , V 2 2 (V 1Ṽ2 − V 1 V 2 ) = V 2 2 I 2 and (V 1Ṽ2 −Ṽ 1 V 2 ) 2 = I 2 2 , respectively. This implies the following normal form: 5 (3.12) Because the homological operator H is nondiagonal, the expressions for the coefficients A j are quite complicated, so we do not provide corresponding formulas.
Consider now a 1-parameter deformation H ε , ε ∈ (C, 0), of the above Hamiltonian. Under some genericity assumption the quadratic part can be transformed to the following form: In [BaSa] it was proved that (in the case of imaginary eigenvalues) a unique normal form is H = H 2 + f (I 2 , V 2 2 ), where f is a formal power series. In [Mar,Ch. 4,Sect. 4] a slightly different normal form is given.
(In the normal form (3.13) we have a 2 = μ 2 + ε and a 4 = μ 2 − ε 2 .) Indeed, then system (3.12) becomes perturbed to Assume ε = 0. In the variables 14) the latter system is diagonalizable with the corresponding eigenvalues We note also that Like in the previous section, we find that the above reduction of cubic and quartic terms for ε = 0 can be extended to the case ε = 0 (but small). We arrive at the following proposition.
Proposition 2. The family H ε can be reduced to the following normal form: ε + A 1 V 2 4 + A 2 V 2 2 I 2 + A 3 I 2 2 + · · · , where the constants A j = A j (ε) depend analytically on the parameter.