Poisson and symplectic reductions of 4-DOF isotropic oscillators. The van der Waals system as benchmark

This paper is devoted to studying Hamiltonian oscillators in 1:1:1:1 resonance with symmetries, which include several models of perturbed Keplerian systems. Normal forms are computed in Poisson and symplectic formalisms, by mean of invariants and Lie-transforms respectively. The first procedure relies on the quadratic invariants associated to the symmetries, and is carried out using Gr\"obner bases. In the symplectic approach, hinging on the maximally superintegrable character of the isotropic oscillator, the normal form is computed {\it a la} Delaunay, using a generalization of those variables for 4-DOF systems. Due to the symmetries of the system, isolated as well as circles of stationary points and invariant tori should be expected. These solutions manifest themselves rather differently in both approaches, due to the constraints among the invariants versus the singularities associated to the Delaunay chart. Taking the generalized van der Waals family as a benchmark, the explicit expression of the Delaunay normalized Hamiltonian up to the second order is presented, showing that it may be extended to higher orders in a straightforward way. The search for the relative equilibria is used for comparison of their main features of both treatments. The pros and cons are given in detail for some values of the parameter and the integrals.


Introduction
The use of computer algebra systems for normal forms computations is considered at present a routine operation. As a general reference see e.g. Sanders et al. [29] and Meyer et al. [26]. Nevertheless when we deal with special classes of differential equations, like Poisson or Hamiltonians systems which is our case, it is advisable to employ specific transformations as well as tailored variables for those problems [26], mostly connected with the symmetries that those systems might possess. More precisely we are interested in perturbed isotropic oscillators in four dimensions (or perturbed harmonic oscillators in 1:1:1:1 resonance). For those Hamiltonian systems, Sanders et al. explain that the 1:...:1 resonance is one of the more complicated, due to large number of terms in the normal form. This proves to be a key issue in computations of higher order approximations, which is needed in bifurcation analysis for some values of the parameters. For this reason, any strategy developed to reduce the algebra involved in the normal form process, as well as in the subsequent analysis built on it, like relative equilibria and their bifurcations, is something really desirable. As we will see, we have to confront with systems of polynomial equations with parameters, which represent a real challenge, even with computer algebra assistance.
Continuing previous work [6,22,23] [22] in the 1:1:1 resonance, and also [17,18,14] in relation to 1:1:1:1 resonance, we consider in R 4 ×R 4 , the symplectic form w = dQ ∧ dq and a Hamiltonian function where defines the isotropic oscillator, with ω a positive constant and ε is an small parameter. In the first part of the paper we simplify expressions assuming ω = 1. The function P is called the perturbation, where β is a parameter vector, which may include also ε. Moreover we consider that systems defined by Hamiltonian function Eq. (1) have two first integrals in involution given by associated to which we have rotational symmetries. We use the same notation as in [18]. Although some of the methods and techniques considered in the paper may be applied to a large family of Hamiltonian systems defined by (1), to give details of those processes we focus on the uniparametric family of Hamiltonian systems defined by where H 6 (Q, q; β) = q 2 1 + q 2 2 + q 2 3 + q 2 4 × β 2 q 2 1 + q 2 2 − q 2 3 − q 2 4 2 + 4 q 2 1 + q 2 2 q 2 3 + q 2 4 (5) where β is now a real parameter. For β = 1 we have a central potential, i.e. an integrable system. When the system restricts to the manifold Ξ = 0 then it is equivalent to the model for the hydrogen atom subject to a generalized van der Waals potential. For β = 0 this system reduces to the model for the quadratic Zeeman effect. When β = √ 2 we have the Van der Waals system. For this reason we have named the system as the generalized Van der Waals 4-D oscillator. A search (see [14]) for some special solutions of the Hamiltonian system defined by the function (4) reveals that there are invariant 2-tori associated to the rotational integrals (3), which include straight-line orbits over the configuration space.
Normalization and reduction using invariants and Lie-transforms are approached in both Poisson and symplectic formalisms. The first procedure (see [17], [5]) relies on the quadratic invariants associated to the symmetries, and is based on previous work of the authors. Enlarging previous studies on the 1:1:1 resonance [6,22], the Hamiltonian (4) is put into normal form with respect to H 2 . Considering the truncated normal form a system is obtained that is invariant under the the S 1 -actions corresponding to H 2 , Ξ, and L 1 . These three actions together generate a T 3 -action. Reduction with respect to it leads to a one-degree-of-freedom system.
Hinging on the maximally superintegrable character of the isotropic oscillator, the symplectic reduction is carried out a la Delaunay using a generalization of those variables to 4-DOF recently proposed in [21]. Based on the normalized equations, our studies on this system focus on relative equilibria and their evolution with the parameter β. Although first order normalization (averaging) is quite generic, the fact that the family includes some integrable cases, makes it necessary to perform higher order normalization for the stability analysis of those cases.
The paper is organized as follows. Section 2 summarizes part of the work done in [15], the three steps of the Poisson reduction are presented with the invariant functions involved, as well as the corresponding reduced Hamiltonian of our model at each step. In Section 3 we proceed likewise carrying out the reduction, although in different order, by three symplectic transformations: Projective Euler and Andoyer as well as 4-D Delaunay. They give again, but now in the symplectic frame, the 1-DOF thrice reduced Hamiltonian system in the open domain where those charts are defined. Once carried out the toral reduction by the symplectic variables in Section 3, Section 4 focuses on the computation of the normal form by Lie transform which, taking advantage of the periodic character of the unperturbed flow, which allows it to solve the homological equation just by quadratures. Finally Section 5 gathers the analysis on relative equilibria which enlarges the results presented in previous papers, in particular the search for periodic orbits.

Poisson approach. Normalization and reduction
There is a large literature which develops both theoretical and computational aspects of normal forms, see for example [2,25,29,26]. In this section we briefly describe several normal forms for first order of system (4) for ω = 1 thoroughly presented in [15]. In such a work, we used Gröbner Bases as main computational tool and, as software, the computer algebra system Maple.
A constructive geometric reduction in stages, including regular and singular reductions, is performed in [15] and only the final steps are presented here. In the Poisson approach the first reduction is done with respect to the H 2 symmetry. This is a regular reduction and the reduced phase space is homeomorphic to CP 3 (see [5]). Then we carry out a second reduction whose resulting orbit space is stratified: four dimensional isomorphic to S 2 ×S 2 , and two singular strata isomorphic to S 2 . Finally we make a third reduction, with the integral L 1 , reducing the system to a 1-DOF system on the thrice reduced phase space. Depending on the relative value of the integrals, they are isomorphic or homomorphic to a 2-sphere, containing one or two singular points. Besides there are singular reduced phase spaces consisting of a single point (see Figure 2 and [18]).
However, in the following we will not use the invariants π i ; instead, we rely on (K i , L j , J k ) invariants introduced in Egea [17] by the following change of coordinates, The first integrals (see Eq. 3) are now among the invariants defining the image. The first order normal form in these invariants is The reduction of the H 2 action may now be performed through the orbit map ρ K,L,J : R 8 → R 16 ; (q, Q) → (H 2 , · · · , J 8 ) .
Note that on the orbit space we have the reduced symmetries due to the reduced actions given by the reduced flows of X Ξ and X L 1 .
2.2 Toroidal reduction over CP 3 with respect to the rotational symmetry Ξ Let ρ be the S 1 -action generated by the Poisson flow of Ξ over CP 3 . The functions are ρ-invariants over CP 3 . This, in turn, leads us to the orbit mapping The orbit space ρ 2 (CP 3 ) is defined as a six dimensional algebraic variety in R 8 by the following relations The reduced phase spaces are obtained by setting H 2 = n, Ξ = ξ and then the second reduced space is isomorphic to S 2 n+ξ × S 2 n−ξ (see Fig. 1). When ξ = 0, it To further reduce from S 2 n+ξ × S 2 n−ξ to V n ζ l , one divides out the S 1 -action, ρ 2 , generated by the Poisson flow defined by L 1 = π 16 − π 11 over S 2 n+ξ × S 2 n−ξ . The 8 invariants for the L 1 action on R 8 are There are 2 + 3 relations defining the third reduced phase space The Poisson structure of the variables S, K and N is given in Table 1.
For more details see [15]. Then, we have defined the orbit map  When we fix a value of L 1 = l, relations (12) define the thrice reduced space which is a surface of revolution, obtained by rotating f (K) around the axis K, . Thus the shape of the reduced phase space, given in Fig. 2, is determined by the positive part of f (K). Since the roots are So f (K) is positive (or zero) in the subsequent intervals of K: The Hamiltonian on the third reduced phase space is Note that the reduced phase spaces as well as the Hamiltonian are invariant under the discrete symmetry S → −S. To see how to exploit it to obtain a full reducction see [7], [16] and [24]. We choose not to further reduce our reduced phase space with respect to these discrete symmetries, because the three dimensional picture makes it easy to access information about the reduced orbits, and in this way one does not introduce additional critical points (fixed points) which need special attention.
In (K, N, S)-space the energy surfaces are parabolic cylinders. Notice that for β 2 = 2/3, the function H is linear in the variable space (K, N, S). Likewise for β 2 = 1, H modulo constants is independent of ξ and l. Moreover for β 2 = 4, H is only a function of K. Since (V n ξ l , {·, ·} 3 , H Ξ,L 1 ) is a Lie-Poisson system, the corresponding dynamics is given by This system can be integrated by means of elliptic functions, but after a classification of the different types of flows is made, as functions of the integrals and the parameter. Only then we will be ready for the integration of a specific initial value problem. Note that in the search of relative equilibria at the thrice reduced level, the geometry involved is really helpful; the intersection of the Hamiltonian surfaces with the reduced phase space gives the trajectories of the reduced system. Then, tangency of the Hamiltonian surface with the reduced phase spaces provides relative equilibria that generically correspond to three dimensional tori in the original phase space. Details on the analysis of this system, focusing on relative equilibria related to β 2 = 0, ξ = l and the case of physical interest ξ = 0, are contained in Díaz et al. [15]. The reader should take into account that β 2 is renamed in [15] as λ.

Symplectic charts for 4-D isotropic oscillators
For our study of perturbed oscillators, we switch now from Poisson to symplectic formulation. More precisely, our goal is to obtain the normal form of isotropic oscillators by Lie transforms, in the presence of symmetries. Different charts are used for the symplectic treatment of the harmonics oscillators. Among them we find complex notation [5], action-angle variables of Poincaré type [29], [23], or variations of them in the case of resonances [13]. In our case, as we also need to reduce by the actions associated to the rotational symmetries, we have made use of a symplectic chart which incorporates the functions (3) as momenta. Moreover the connection between the isotropic and Kepler problems, two maximally superintegrable systems, allows to introduce the Delaunay transformation [11]. The result is the 4-D Delaunay transformation proposed by one of the authors [21], that we will use in what follows. There is an alternative procedure considering the Lissajous transformation [12], which will be presented elsewhere.

The family of Euler projective transformations
Let F (ρ) be a smooth real function which is positive in its domain. We consider the family of transformations: PE F : (ρ, φ, θ, ψ) → (q 1 , q 2 , q 3 , q 4 ), dubbed as Projective Euler variables, given by For F (ρ) = 1, this transformation defines Euler parameters as functions of Euler angles. In this paper, we only consider the case F (ρ) = √ ρ. The canonical extension associated to the transformation (15) is readily obtained as a Mathieu transformation, which satisfies Q i dq i = P dρ + Θ dθ + Ψ dψ + Φ dφ. The relations among the momenta are given by Later on we will need the inverse transformation given by , .
(18) Proof : Considering any given initial condition for the Hamiltonian H ω = h, by applying relations (15) and (16), we obtain, by straight forward computations, the above expression for the isotropic oscillator H 2 . q.e.d.
Note that the Hamiltonian has two cyclic variables, which manifests the two symmetries associated with our system we have refer in (3)
Theorem 3.2 In Projective Andoyer variables, the system defined by (2), regularized by ds = 1/(4ρ) dτ , includes the Keplerian system for any value of the integral U 3 . Proof : Following Poincaré technique, we first regularize (2) and our new Hamiltonian will be K = (H ω − h)/(4ρ). Considering now Proposition 3.1, after some straightforward calculations, the Hamiltonian of the 4-D isotropic oscillator is given byK in the manifoldK = −ω/8 and γ = h 4 . But the above Hamiltonian corresponds to the Kepler system in polar nodal variables, see [10]. q.e.d.

Delaunay symplectic chart
Following [11] we plan to normalize perturbed isotropic systems by Lie-transforms a la Delaunay in the following Section. Thus, we still need to implement (in part of the phase space) another canonical transformation of Hamilton-Jacobi type: D γ : (L, G, ℓ, g) → (ρ, u 2 , P, U 2 ). The process hinges on the H-J equation built on the function (21). For details of this transformation D γ we refer to [11]; here we only include the final expressions. Among the relations defining Delaunay transformation, which can only be given in implicit form, we take where a = a(L), e = e(L, G) are given by and f and E are auxiliary angles. The symplectic variable ℓ is related to them by Later on we also need 1/ρ = (1 − e cos f )/p, where p = aη 2 , and Completing the functions of the momenta given above (23), it is convenient to introduce two more state functions The previous results may be summarized in the following theorem: Theorem 3.3 By composing the three symplectic transformations together with the regularization ds = 1/(4ρ) dτ , the Hamiltonian of the 4-D isotropic oscillator is given by in the manifold H 0 = −ω/8.
The set of variables (ℓ, g, u 1 , u 3 , L, G, U 1 , U 3 ) is what we call 4-D Delaunay chart. As in 3-D with the classical Delaunay variables, they represent a generalized set of action-angle variables (see [28]). In the next section we show the interest of this symplectic transformation. For the benefit of the reader let us mention that Moser and Zehnder [27] introduced action-angle variables for the Kepler problem in R n and called them Delaunay variables. When we restrict to n = 4, those variables do not coincide with the set built in this paper. Finally, in order to compare results of our analysis of relative equilibria, we need to establish the connection between the variables K (Poisson approach) and G (symplectic approach) defining the thrice reduced space. After some algebraic manipulations, we obtain which expresses G as function of K and N , invariants defining the thrice reduced space, and the first integrals.

Delaunay normalization of perturbed 4-D isotropic oscillators
Our goal in this section is to make normalization by Lie transforms [9], using 4-D Delaunay variables. As we will see they allow to express perturbed isotropic oscillators in a 'convenient' form, to implement Lie transforms up to higher order in an efficient way. What we mean by convenient is the following. It is well known that the isotropic oscillator H 0 , as the Kepler system, defines a maximally superintegrable Hamiltonian system (see Fassò [20]), whose flow is made of periodic orbits. In that case Cushman [3] proved that any smooth function F over its phase space may be decomposed in a unique way into a sum (ii) There exists a smooth functionF such that {F , H 0 } = F * ; in other words F * belongs to the image of the operator L 0 . In particular when the Lie transform (see [9], [26]) is carried out a la Delaunay [11], the homological equation, which relates the new Hamiltonian (ǫ j /j!)H ′ j with the generating function (ǫ j /j!)W ′ j , is given by Then, considering the splitting coming from Cushman theorem, the equation (29) may be solved choosing where, according with (27), ∂H 0 /∂L = γ 2 /L 3 . The Hamiltonian (1) of perturbed isotropic oscillators in Delaunay variables is given by and the normalized Hamiltonian up to order k takes the form The main feature of the process is that, at each order, solving the homological equation (29) only involves quadratures.

Delaunay normalization and symmetries
Apart from the procedure associated to Delaunay normalization we have just mentioned, when we restrict to systems with the symmetries (3), the use of Delaunay chart shows its full advantage. Indeed, in that case u 1 and u 3 are cyclic, in other words, we have In geometric language, the use of Delaunay variables has carried out the toral reduction associated to the actions defined by the symmetries (3), and the function (31) is the reduced Hamiltonian. The Hamiltonian system given by (31) is a 2-DOF system. Thus, the normalized system will take the form In our case normalizing up to the order needed, after truncating, we will obtain an integrable 1-DOF Hamiltonian system. From now on we drop primes in the variables in order to simplify the expressions.

Carrying out the normalization. The van der Waals model as a benchmark
In order to implement these normalizations, since we are dealing with polynomial perturbations, from experience with 3-DOF Kepler systems we introduce the auxiliary variable E by using the following proposition.  (20), (22) and (25). q.e.d.
Applying the above result, the Hamiltonian (4) takes the form where functions C ij = C ij (g, L, G, U 1 , U 3 ; β) and S ij = S ij (g, L, G, U 1 , U 3 ; β) belong to the kernel of the Lie transform. Thus, by considering the differential relation (25) and by replacing in (30), we obtain the part in the kernel as follows Then, we solve the first order homological equation (29). The first order normalized Hamiltonian takes the form The first order W 1 of the generating function is obtained computing the second quadrature in (30), then, taking into account the shorthand α = β 2 − 1, we get the following expression c 1 e 3 sin(3u) − 9e 2 sin(2E) + 3e(3η 2 + 5) sin(u) + αs 1 s 2 +3(−2η 3 + η 2 + 5) sin(g + E) .
Analogously we obtain the second order normalization for (32), which will be used in the study of stability. The second order coefficients are where Λ n,i = c 2 1 + c 2 2 (−1) i n c 2 1 c 2 2 (−1) i . As the van der Waals system includes several integrable cases, in order to study stability in those cases, higher order normalization, at least second order, is needed. This subject will be tackled in future works.
Here we try only to identify relative equilibria. In fact, this is the procedure to follow in order to classify and compute different types of orbits defined by our system. For the generic case, first order normalization is sufficient. The second order normalization is required when we consider degenerate situations, connected with integrable cases. For this reason we give both, first and second order in Section 4.2.
5 Searching for relative equilibria and invariant tori of the first order normalized system Focusing on the task that we have announced before, we use the first order normalization of the Hamiltonian. Thus the corresponding differential system is given byl In other words L, U 1 and U 3 are integrals, as we already know, and the system splits into a 1-DOF Hamiltonian system, namelẏ and three quadratures coming froṁ as soon as the system Eqs. (39) is solved. We are not going to explore this way, which moreover involves hyperelliptic integrals. In other words, it will be more convenient to rely on numerical integrations. As we have already said, our previous work on this model concentrated on searching for relative equilibria related to singular points of the energy-moment map. We have already mentioned that the complete analysis should be done using invariants. In fact the lower dimensional relative equilibria, i.e. those that correspond to invariant S 1 or T 2 , are given by the singularity of the moment map for the T 3 -symmetry group, and can be described by a moment polytope; for details see [15].
Excluding those solutions, in the remaining open domain we may use symplectic charts. Relative equilibria of the system (35) -(38) may be classified as follows: • invariant 3-tori. They are the solutions (g, G) of the system defined bẏ as functions of U 1 , U 3 and β.
• invariant 2-tori. In a similar way, the search for invariant 2-tori splits into the search the roots of the subsystems. Namelẏ where the possible roots will be function of U 3 and β, anḋ where the possible roots will be function of U 1 and β.
• periodic orbits: The research for periodic orbits of the original system is equivalent [30] to find the roots of the first reduced system of equationṡ Solutions of the system given by Eqs. (40) define invariant 3-tori T 3 (ℓ, u 1 , u 3 ), except for the singular points [15]. The system defined by Eqs. (41) gives the invariant 2-tori T 2 (ℓ, u 3 ), each of them is attached to a 2-torus T 2 (g, u 1 ) made of fixed points. Likewise the system defined by Eqs. (42) is the invariant 2-tori T 2 (ℓ, u 1 ), each of them is attached to a 2-torus T 2 (g, u 3 ) made of fixed points. Finally, the intersection of the previous types of 2-tori, defined by the system of Eqs. (43), correspond to periodic solutions S 1 (ℓ), each of them is attached to a 3-torus T 3 (g, u 1 , u 3 ) made of fixed points. They correspond to the short period solutions related to the unperturbed system. In what follows we search for invariant 3-tori and periodic solutions of short period.
The search for possible invariant T 2 solutions of Systems (41) and (42) requires a similar analysis. We will not refer to them here. Instead, we focus on periodic orbits, where we obtain a benefit working in symplectic formalism.

Searching for periodic orbits
With Poisson formalism, finding periodic solutions is in correspondence with computing relative equilibria of the system defined with the normalized Hamiltonian (9). This study requires to deal with the constraints defining CP 3 and leads to a large polynomial system; for details see [15]. Although partial results are obtained, the presence of the physical parameter introduces complicated expressions.
Thus, we have either c 2 = 0 or Thus, for each member of the Van der Waals family defined by α, we have the relation between integrals which give periodic orbits. The period is computed replacing those values in the right hand terms of (35).

Conclusions and future work
In contrast to some claims in [4], Poisson and symplectic formulations of perturbed 4-D isotropic oscillators systems behave as complementary in the analysis of the main features of those systems. The first approach, built on the invariants and their algebraic relations, is necessary in order to carry out regular and singular reductions of the systems, in particular when singular points are present in the reduced spaces. Nevertheless the use of those invariants and their relations, introduce large computations in the search of relative equilibria, specially when physical parameters are involved. Only those equilibria related to the symmetry groups of the system are more easily computable. The situation is rather different when the symplectic approach is followed. From the geometric mechanics point of view this helps to portrait the toral structure of the phase space, necessary when stability KAM theory is applied. Moreover, from the algebraic perspective, the number of equations and relations involved falls sharply because there are no constraints. Normal forms are computed very efficiently in this frame. Of course, this is at the expense of considering only an open domain of the phase space, where those symplectic variables are defined. For this reason we should begin with Poisson formalism. It is only after studding singular points that we can switch to symplectic techniques with properly chosen variables.
As an illustration, the Van der Waals family is studied in detail here, showing the pros and cons of both approaches. The reconstruction process connecting the whole analysis of relative equilibria with the original system, is still to be tackled. In particular, the analysis of the different types periodic orbits.
An open question is whether there are other invariants which can lead to similar equations to those appearing in symplectic variables . In the same vein, connected with our choice as 4D-Delaunay variables [23], it might be worth considering other recent proposals [17], [30]. Results on this, now in progress, will be published in [3].