Periodic solutions of symmetric Hamiltonian systems

This paper is devoted to the study of periodic solutions of Hamiltonian system ż(t) = J∇H(z(t)), where H is symmetric under an action of a compact Lie group. We are looking for periodic solutions in a nearby of non-isolated critical points of H which form orbits of the group action. We prove Lyapunov-type theorem for symmetric Hamiltonian systems.


Introduction
Consider a first-order systemż (t) = J∇H(z(t)) (HS) on R 2N , where J = 0 I −I 0 is the standard symplectic matrix and H : R 2N → R is a Hamiltonian of the class C 2 . The existence of periodic orbits in Hamiltonian dynamics is an important and widely studied problem. In 1895 Lyapunov [17] proved his center theorem i.e. the existence of one-parameter family of periodic solutions of (HS) tending to a non-degenerate equilibrium. The next important result of Weinstein [31] shows the existence of at least N geometrically distinct periodic solutions at any energy level of the Hamiltonian H. The further development of Weinstein theorem was performed by Moser [20]. In 1978 Fadell and Rabinowitz proved the lower bound for the number of small nontrivial solutions of (HS) depending on the period. See [24] for the general overview of the results up to 1982. The results of Weinstein and Moser were generalized by Bartsch in 1997, [2]. The problem of the existence of periodic solutions of (HS) in a case of degenerate equilibrium was also studied by Szulkin [29] and Dancer with Rybicki [5] who generalized classical result of Lyapunov. Suppose now that the compact Lie group Γ acts unitary on R 2N and H ∈ C 2 (R 2N , R) is a Γ-invariant potential i.e. H(γz) = H(z) for any γ ∈ Γ and z ∈ R 2N . The study of the existence of periodic solutions in this case was performed by Montaldi, Roberts and Stewart [19] who had proved equivariant version of Weinstein-Moser theorem. In 1993 Bartsch [1] generalized the theorem of Montaldi, Roberts and Stewart for the wider class of a group actions which allows him to generalize the result of Fadell and Rabinowitz also. However, the authors mentioned above have assumed that critical point z 0 of H is a fixed point of group action i.e. the orbit of action consists of one point. Then z 0 can be an isolated critical point.
We study a more general case. Assume that z 0 is a critical point of Hamiltonian H. Since H is Γ-invariant, Γ(z 0 ) = {γz 0 : γ ∈ Γ} ⊂ (∇H) −1 (0), i.e. the orbit Γ(z 0 ) consists of critical points of H and, therefore, stationary solutions of the equation (HS), see Remark 2.4. Hence, if dim Γ z 0 < dim Γ then the orbit is at least an one-dimensional manifold and, as a consequence, critical points are not isolated. Therefore the results mentioned above are not applicable. We are going to prove sufficient conditions for the existence of non-constant periodic solutions of an autonomous Hamiltonian system in the presence of symmetries of a compact Lie group, the problem (HS), in any neighborhood of the orbit Γ(z 0 ), see the main result Theorem 4.1 and Theorems 5. 2, 5.3, 5.4. This article is organized as follows. In section 2 we recall some basic definitions and notions of group theory and equivariant topology. Equivariant Conley index which is a main tool of our reasoning is shortly defined in subsection 2.3. Furhter, we recall the notion of Euler ring, equivariant Euler characteristic and its generalization, see Remarks 2.7, 2.8. In Theorem 2.11 we recall the very important theorem connecting equivariant Euler characteristic, equivariant Conley index and the idea of orthogonal section introduced in the paper [21]. In subsection 2.5 we formulate so called equivariant splitting lemma -the theorem which allows us to simplify the study of Conley indexes up to the linear case in Lemma 4.8.
In section 3 we parameterize the equation (HS) to study the solutions with constant period 2π in the equation (HS-P). Next we introduce appropriate Sobolev space E, the action of the group Γ × S 1 on it and we define variational functional Φ : E → R (see formulas (3.4), (3.5)) such that 2π-periodic solutions of the system (HS-P) are in bijective correspondence with S 1 -orbits of critical points of Φ. In this way we begin to study the equation (3.7). Further, we analyze the linear Hamiltonian system (HS-L); it is a base for the last step in the proof of the main result of the paper. Section 4 is devoted to the formulation and the proof of the main result of this paper, Theorem 4.1. The notion of bifurcation theory is recalled in Definition 4.3 and in the nearby text. In Theorems 4.5, 4.6 we formulate the necessary and sufficient condition for the existence of global bifurcation of solutions of the equation (3.7). The last part of this section is devoted to the proof of the change of equivariant Euler characteristics of equivariant Conley indexes i.e. the formula (4.1). Firstly, we reduce our task to the space orthogonal to the orbit, see Lemma 4.7 and the text above them. Next in Lemma 4.8 we reduce the problem to the linear case. To study it we prove Theorem 4.10. To finish the proof of the main result we study the minimal periods and convergence of new solutions in Remarks 4.11,4.12. In the fifth section we reformulate the main result to make the assumptions easier to verify. The most friendly version of our result is the following theorem (see Theorem 5.4).
, 0) = 0 for sufficiently small ǫ and m + (∇ 2 H(z 0 )) = N . Then there exists a connected family of non-stationary periodic solutions of the systemż(t) = J∇H(z(t)) emanating from the stationary solution z 0 such that periods (not necessarily minimal) of solutions in the small neighborhood of z 0 are close to 2π/β j , where iβ j , β j > 0, is some eigenvalue of J∇ 2 H(z 0 ).
For the two other versions see Theorems 5.2 and 5.3.
Furhter, we show that Lyapunov-type theorem of Dancer and Rybicki (Theorem 5.5) is generalized by the main result of this paper -Theorem 4.1. In the last part of this section we reformulate the second-order Newtonian system to the Hamiltonian one. Then the two symmetric versions of the Lyapunov center theorem, Theorem 5.6 proven in [21] and Theorem 5.7 proven in [22], are also the consequences of the results proven in this paper.
The last section is devoted to an interesting application of the abstract results of this paper. We study the existence of quasi-periodic motions of the satellite in a nearby of a geostationary orbit of an oblate spheroid. In order to do this we consider a gravitational motion in the rotating frame where the corresponding Hamiltonian is given by formula (6.2). It is SO(2) invariant and possesses a critical point which represents the geostationary orbit in the original coordinates. Theorem 5.4 will be directly applied in this problem to prove the existence of trajectories with arbitrarily small deviations from the geostationary ones.

Preliminaries
In this section we recall the basic material on equivariant topology from [30,14] and prove some preliminary results. Throughout this section G stands for a compact Lie group.
2.1. Groups and their representations. Denote by sub(G) the set of all closed subgroups of G. We say that two subgroups H, The conjugacy is an equivalence relation on sub(G). The class of H ∈ sub(G) will be denoted by (H) G and the set of conjugacy classes we denote by sub [G].
If x ∈ R n then G(x) = {gx : g ∈ G} is the orbit through x and a group G x = {g ∈ G : gx = x} ∈ sub(G) is called the isotropy group of x. The isotropy groups of the elements of common orbit are conjugate i.e if G( Below we recall the notion of an admissible pair, which was introduced in [21], where one can find some examples and properties. Definition 2.1. Fix H ∈ sub(G). A pair (G, H) is said to be admissible if for any K 1 , K 2 ∈ sub(H) the following condition is satisfied: Note that if Γ is a compact Lie group, then the pair (Γ×S 1 , {e}×S 1 ) is admissible, see Lemma 2.1 of [21]. This property will play a crucial role in the proof of the main result, Theorem 4.1.
Recall that a unitary group U (N ) id defined by is a symplectic group and we call a real, unitary representation of G. To simplify notation we write gx instead of ρ(g)x and R 2N instead of V if the homomorphism is known.
Two  Definition 2.2. We say φ : Ω → R of class C k is G-invariant C k -potential, if φ(gx) = φ(x) for every g ∈ G and x ∈ Ω. The set of G-invariant C k -potentials will be denoted by C k G (Ω, R). Definition 2.3. A map ψ : Ω → V of the class C k−1 is called G-equivariant C k−1 -map, if ψ(gx) = gψ(x) for every g ∈ G and x ∈ Ω. Then we write ψ ∈ C k−1 G (Ω, V). Fix ϕ ∈ C 2 G (Ω, R). By ∇ϕ, ∇ 2 ϕ we denote the gradient and the Hessian of ϕ, respectively.
critical points form orbits of a group action. If ∇ϕ(x 0 ) = 0 then ∇ϕ(·) is fixed on G(x 0 ) and therefore

Equivariant Conley index.
In this section we shortly recall the construction of equivariant Conley index introduced by Izydorek [13], see also [8,26].
Denote by F * (G) the category of finite pointed G-CW-complexes (see [30] for definition and examples). The G-homotopy type of X ∈ F * (G) we denote by [X] G ∈ F * [G] (or [X] when no confusion can arise) and by F * [G] the set of G-homotopy types of elements of F * (G). If X is a G-CW-complex without a base point, then we denote by X + a pointed G-CW-complex X + = X ∪ { * }. By CI G (S, ϑ) we denote a finite dimensional G-equivariant Conley index of an isolated invariant set S under a G-equivariant vector field ϑ, see [1,6,7,27] for the definition. Recall that CI G (S, ϑ) ∈ F * [G].
Let ξ = (V n ) ∞ n=0 be a sequence of finite-dimensional orthogonal G-representations.
The set of G-spectra of type ξ is denoted by GS(ξ). We can also define G-homotopy equivalence of two spectra E(ξ), E ′ (ξ) (see [13] for the details). The G-homotopy type of a G-spectrum E(ξ) we denote by [E(ξ)] G (or shorter [E(ξ)]) and the set of G-homotopy types of G-spectra by [GS(ξ)] or simply [GS] when ξ is fixed.
Let (H, ·, · ) be an infinite-dimensional orthogonal Hilbert representation of a compact Lie group G.  [13], generated by ∇Φ. Let be O an isolating G-neighborhood for ϑ and put N = Inv ϑ O. Set ξ = (H + k ) ∞ k=1 . Let Φ n : H n → R be given by Φ n = Φ |H n and ϑ n denotes the G-flow generated by ∇Φ n . Note that ∇Φ n (x) = Lx+P n •∇K(x). Choose n 0 such that for n ≥ n 0 the set O n := O ∩ H n is an isolating G-neighborhood for the flow ϑ n . Then the set Inv ϑn (O n ) admits a G-equivariantindex pair (Y n , Z n ).
We define a spectrum E(ξ) . Since isolated invariant set O is defined by isolating neighborhood N and the flow is related to vector field ∇Φ we will also write CI G (N , ∇Φ).

Equivariant Euler characteristic.
Let (U (G), +, ⋆)) be the Euler ring of G, see [30] for the definition and more details. Let us briefly recall that the Euler ring U (G) is commutative, generated by is the universal additive invariant for finite pointed G-CW-complexes known as the equivariant Euler characteristic.
Remark 2.7. Below we present some properties of the Euler characteristic χ G (·).
• For X, Y ∈ F * (G) we have: For the prove of this fact see Lemma 3.4 in [15].
There is a natural extension of the equivariant Euler characteristic for finite pointed G-CWcomplexes to the category of G-equivariant spectra due to Gołȩbiewska and Rybicki [11].
Recall that due to Remark 2.7 an element χ G (S V n ) is invertible in the Euler ring U (G) and define a map Υ G : [GS(ξ)] → U (G) by the following formula Theorem 2.10. If N 1 , N 2 are isolated G-invariant sets for the local G-LS flows generated by ∇Ψ 1 and ∇Ψ 2 respectively then The following theorem is one of the most important fact in our reasoning. It allows us to simplify the distinguishing of the infinite-dimensional equivariant Conley indexes, significantly.
Let H = ∞ n=0 H n be a representation of the compact Lie group G. Consider two functionals be isolated orbits of critical points of the potentials ϕ 1 and ϕ 2 , respectively. Moreover, assume that G The proof of the theorem above is based on a concept of smash product over group. One can find more details in [22], especially Definition 2.4.2, Theorem 2.4.1 and Theorem 2.4.2.
2.5. Equivariant splitting lemma. Let K be a compact Lie group and let (V, ·, · ) be an orthogonal Hilbert representation of K with an invariant scalar product ·, · . Moreover, assume Consider a functional Ψ ∈ C 2 K (Ω, R)of the form which satisfies the following assumptions Note that the kernel ker A and the image im A are orthogonal representations of K. Moreover, ker A is finite dimensional and trivial representation of The following theorem (called splitting lemma) provides the existence of equivariant homotopy which allows us to study the product (splitted) flow . The proof of this theorem one can find in [22] (Theorem 2.5.2).
3) and satisfies assumptions (F.1)-(F.5). Then, there exists ε 0 > 0 and K-equivariant gradient homotopy ∇H : (B ε 0 (ker A) × B ε 0 (im A)) × [0, 1] → V satisfying the following conditions: (4) There exists an K-equivariant, gradient mapping ∇ϕ : The homotopy H is given by Moreover, from the proof of Theorem 2.5.1 follows that the potential ϕ : Remark 2.5.3. Note that we don't assume that ker A = {0}. In the case of trivial kernel the homotopy given in Theorem 2.5.1 provides a linearization of functional.

Variational formulation for Hamiltonian systems
Recall that we are interested in the existence of periodic solutions with any period of the system (HS). In order to find them we are going to study 2π-periodic solutions of the parameterized systemż (t) = λJ∇H(z(t)), (HS-P) which are in one-to-one correspondence to 2πλ-periodic solutions of the system (HS).
To prove the existence of solutions of the Hamiltonian system (HS-P) we are going to the study critical points of a corresponding functional.
Define the Sobolev space of 2π-periodic R 2N -valued functions where · denotes the standard scalar product, is a Hilbert space usually denoted by Since we consider R 2N as a unitary representation of the compact Lie group Γ, E is a unitary G = Γ × S 1 -representation with the action given by and therefore the action proposed in (3.2) is given on E k by the product of unitary matrices γ 0 0 γ and Remark 3.1. Since we are going to study the Hamiltonian system (HS-P) is a neighborhood of the the orbit G(z 0 ) of critical points, without loss of generality we can assume that Hamiltonian H satisfies the following growth restriction Indeed, we may chooseH such that ∇H is bounded (i.e. s = 1) andH(z) = H(z) in a neighborhood of the orbit G(z 0 ).
It is known (see [18]) that periodic solutions of the system (HS-P) are in one to one correspondence with S 1 -orbits of critical points of a potential Φ : see [9], the formula (3.3).
Since we consider R 2N as a unitary representation of a group Γ and H is Γ-invariant, the potential Φ is Γ-invariant. Moreover, it is S 1 -invariant since it acts on 2π-periodic functions.
Recall that since the Hamiltonian H is Γ-invariant, the solutions of the system (HS-P) form Γ-orbits i.e. if z 0 is a solution on (HS-P) then γz 0 solves (HS-P) for any γ ∈ Γ. Therefore we are going to study G = Γ × S 1 -orbits of critical points of the corresponding G-invariant potential Φ i.e. we are interested in solutions of the system ∇ z Φ(z, λ) = 0. (3.7) Note that ∇ z Φ(z, λ) = Lz + ∇K λ (z), L is a linear, self-adjoint and G-equivariant operator and ∇K λ (z) is completely continuous. Since ker L = E 0 and L |E ± k = ±Id, the conditions (B.1)-(B.3) given on the page 4 are satisfied.
Let z 0 ∈ (∇H) −1 (0) and consider a linear Hamiltonian systeṁ which has a form of (HS-P) with H(z) (3.8) Taking into account the scalar product in E given by (3.1) and the formula (3.8) we obtain and as a consequence It means that ∇Φ L (z, λ) acts on E k = {a cos(kt) + b sin(kt) : a, b ∈ R 2N } for k ≥ 1 as a linear map It is easy to see that equation

Main Result
In this section we prove our main result of this paper i.e. the global bifurcation of periodic solutions of the system (HS) in the most general version. We emphasize our assumptions (A1) H : R 2N → R is a Γ-invariant Hamiltonian of the class C 2 , (A2) z 0 ∈ R 2N is a critical point of H such that the isotropy group Γ z 0 is trivial,  The assumption (A7) is very general and laborious to verify. We will change and simplify them in some specific cases. However, it does not follow directly from the structure of a Hamiltonian system in general situation as we obtained in a study of Newtonian systems, see [21], the proof of Lemma 4.1.  Put Λ = { k β j : k ∈ N, iβ j ∈ σ(J∇ 2 z H(z 0 ))}. In the theorem below we prove the necessary condition for the existence of bifurcation from the orbit G(z 0 ) × {λ 0 }.
Proof. By a reasoning given in the proof of Theorem 3.2.1 in [22] we obtain ker To complete the proof we have to prove that it implies λ 0 ∈ Λ. The study of the kernel of ∇ 2 z Φ(z 0 , λ 0 ) is equivalent to the study of the linearized system (HS-L) where A = ∇ 2 z H(z 0 ). Therefore by the Lemma 3.2 we obtain the thesis. Choose λ 0 such that the necessary condition and assumptions (A6), (A7) are satisfied i.e. λ 0 = 1 β j 0 ∈ Λ and put λ ± = 1±ε β j 0 such that λ ± > 0 and [λ − , λ + ] ∩ Λ = {λ 0 }. To prove the existence of global bifurcation we are going to apply the following theorem Proof. The theorem above follows directly from the relation [3], Theorem 3.10) and from a global bifurcation theorem for equivariant gradient degree (see [10], Theorem 3.3).
Define H ⊂ E by H = T ⊥ z 0 G(z 0 ). Recall that the space perpendicular to the orbit at z 0 is an G z 0 -representation. Since z 0 is a constant function and by the assumption (A2) G z 0 = {e} × S 1 , H is an unitary S 1 -representation.
Put Ψ ± : H → R by Ψ ± (z) = Φ(z, λ ± ). Note that since H is an S 1 -representation, the potenatial Ψ ± is S 1 -invariant. Moreover, z 0 is an isolated critical point of Ψ ± . Since G(z 0 ) = Γ(z 0 ) ⊂ E 0 we have the following decomposition In order to prove the main result of this paper we prove the existence of global bifurcation from the orbit G(z 0 )×{λ 0 } i.e. we need to prove formula (4.1). In the theorem below we simplify this formula to the study of potentials defined on the orthogonal section T ⊥ z 0 G(z 0 ).

Lemma 4.7. Under the above assumptions if
Proof. Since the pair (Γ × S 1 , {e} × S 1 ) is admissible (because S 1 is abelian), see Definition 2.1 and both ∇Φ(·, λ − ), ∇Φ(·, λ + ) are in the form of a compact perturbation of the same linear operator L, we can apply Theorem 2.11 to obtain the thesis directly.
From now our goal is to prove formula (4.2). The next step is to transform a problem into the study of Conley indexes with simpler structure of flows.
We define H : T ⊥ z 0 Γ(z 0 ) → R by H(z) = H(z + z 0 ) and Ψ ± : H → R by Ψ ± (z) = Ψ ± (z + z 0 ). Since Ψ ±|E 0 = 2πλ ± H and the orbits G(z 0 ) × {λ + }, G(z 0 ) × {λ − } do not satisfy the necessary condition for the existence of bifurcation we obtain ker ∇ 2 Ψ ±| (0) = ker ∇ 2H (0) so the kernel is independent on λ ± . Since ∇ 2Ψ ± (0) is self-adjoint we are able to decompose The next theorem simplifies the proof of formula (4.2) to the study of Conley indexes of linear vector fields. In order to prove it we apply splitting lemma (Theorem 2.5.1).  Proof. It is clear that by the properties of Conley index we have Since we are going to apply splitting lemma (Theorem 2.5.1), now we verify that Ψ ± satisfies conditions (F.1)-(F.5) given on the page 6 with K = S 1 , Moreover, a hessian is a self-adjont operator. By Theorem 4.5 ker 0)z and both summands are compact and S 1 -equivariant, ∇ζ ± is also compact and S 1 -equivariant. (F.4) It is obvious due to formula given in (F.3). (F.5) Since λ ± / ∈ Λ i.e. the orbits G(z 0 ) × {λ ± } do not satisfy the necessary conditions for the existence of bifurcations, the orbit G(z 0 ) is isolated in the set (∇Φ(·, λ ± )) −1 (0). Therefore 0 ∈ H is an isolated critical point ofΨ ± . Applying Theorem 2.5.1 (splitting lemma) and Theorem 2.10 (product formula) we obtain Since R 0 is an invariant space of the linear map ∇ 2 Ψ ± (0) we are able to decompose the linear flow to obtain and combining the flows given on the E 0 ∩ H we finally obtain If we study the homotopy H (see Theorem 2.5.1 and Remark 2.5.2) acting on the subspace of constant function E 0 we obtain By the homotopy invariance of the Conley index and since λ − , λ + are both positive we have . Note that the space E 0 is finite-dimensional and consists of constant functions (elements invariant on S 1 action), therefore for sufficiently small ε > 0, where the last equality follows from Poincaré-Hopf theorem, see [28]. Since we finally have By the assumption (A5) deg(∇H |T ⊥ z 0 Γ(z 0 ) , B(z 0 , ǫ), 0) = 0 and due to equation (4.7) we obtain that the formula (4.2) is equivalent to and the proof is completed.
To verify formula (4.4) we are going to study equivariant Conley index and equivariant Euler characteristic by definitions. Note that the vector field −∇Π ± : R ∞ → R ∞ is linear and the decomposition R ∞ = ∞ k=1 E k satisfies conditions (B.1)-(B.3) given on the page 4. Recall that λ 0 = 1 Remark 4.9. Note that the linearization of the variational functional Φ for the parameterized Hamiltonian system (HS-P) is equal to variational functional for the linearized system (HS-L) (we remove high order tenses in both cases). Therefore the action of the linear vector field for k large enough, say for k ≥ k 0 .
By the assumption (A7) Since E 1 is non-trivial S 1 -representation by Remark 2.7 we obtain Combining formulas (4.11) and (4.12) we finally obtain which completes the proof. . By the non-resonance condition for eigenvalues (i.e β j /β j 0 ∈ N for all j = j 0 ) we obtain λ 0 r / ∈ Λ = { k βr : k ∈ N, iβ r ∈ σ(J∇ 2 z H(z 0 ))} for any r ∈ N and therefore there are no 2πλ 0 r -periodic non-stationary solutions in a neighborhood of the orbit Γ(z 0 ). Hence we can consider periods tending to 2π   R 2N ). Now we prove that corresponding periodic solutions of Hamiltonian system tend to z 0 ∈ R 2N in L ∞ -norm. Let z(t) = a 0 + ∞ k=1 a k cos(kt)+ b k sin(kt) be a solution of (HS-P) for λ close to λ 0 . Firstly, Under the condition (3.3), the map z(t) → ∇H(z(t)) is continuous from L 2 (S 1 ) to L 2 (S 1 ) (see Proposition B.1 in [25]). Let ε > 0 and choose Since z is a solution of (HS-P) we obtain Applying Sobolev inequality (see Proposition 1.1 in [18]) we obtain Since λ is bounded in the neighborhood of λ 0 the convergence of solutions z to z 0 in the norm of H 1 implies the convergence in L ∞ which completes the proof.
Remark 4.13. The assumption (A6) was used only in the proof of Theorem 4.10 i.e. in the last step of the proof of our main theorem. We are able to remove this assumption, but then in the proof of Theorem 4.10 we need to study λ-depending decompositions of not only E 1 but any E k such that 1 β j 0 = k β j for some j. It will cause a complicated notation and the proof will be less readable. However, the change of dim E k,λ,+ when λ varies we will obtain in the same way as for k = 1. Note that assumption (A6) is always satisfied for j 0 = 1 since β 1 is the maximum of β i .

Corollaries
In this section we study in which way is it possible to modify assumption (A7). Moreover, we show that the results of this paper are generalizations of some versions of Lyapunov center theorem.
The following theorem was proven by Szulkin ([29], Proposition 3.6) Theorem 5.1. Suppose that A is symmetric and iβ j , β j > 0, is an eigenvalue of JA. Let E j be the eigenspace of JA in C 2N corresponding to iβ j and Z j the invariant subspace of JA in R 2N corresponding to ±iβ j . Then m − (T 1,λ (A)) changes at λ = 1/β j if and only if the following two equivalent conditions are satisfied: Due to the theorem above we are able to formulate new versions of assumption (A7): where E j be the eigenspace of J∇ 2 H(z 0 ) in C 2N corresponding to iβ j and Z j the invariant subspace of J∇ 2 H(z 0 ) in R 2N corresponding to ±iβ j . Note that if ∇ 2 H(z 0 ) |Z j 0 is a definite matrix then the condition (A6.1) is satisfied. Therefore we put Under this condition the assumption (A7.3) is satisfied for some eigenvalue of J∇ 2 H(z 0 ) and we do not know it precisely. Therefore we exclude assumption (A6). In the theorem below we prove the existence of periodic solutions of the system (HS) without information about their minimal periods. Under the reasoning above, it is clear that Theorem 5.3 is a direct consequence of Theorem 4.1. Looking on the T 1,λ (∇ 2 H(z 0 )) from the other point of view we see Therefore if m + (∇ 2 H(z 0 )) = N then m − (T 1,λ (∇ 2 H(z 0 ))) changes at some λ ∈ (0, ∞). Recall that the levels λ where it can change is Λ (see Lemma 3.2). Therefore the change of m − (T 1,λ (∇ 2 H(z 0 ))) implies the existence of purely imaginary eigenvalue of J∇ 2 H(z 0 ). As a consequence we can propose new condition (A7.5) m + (∇ 2 H(z 0 )) = N and we are able to formulate the next theorem without assumption (A4).
Below we present in which way the theorems presented above generalize classical Lyapunov center theorem and an analogous theorem for Hamiltonian systems that has been proved by Dancer and Rybicki [5]. Moreover, two symmetric version of the Lyapunov center theorem proposed in [21] and [22] are generalized in this paper. Proof. This theorem follows directly from Theorem 4.1 if we consider trivial group Γ = {e}. In this case T z 0 Γ(z 0 ) = R 2N + z 0 .
Consider a Newtonian (second-order) system where U : R N → R is Γ-invariant potential of the class C 2 , Γ acts orthogonally on R N and q 0 ∈ (∇U ) −1 (0). If we substitute r =q we can reformulate the second-order system (NS) to the first-order system q(t) = r(t), r(t) = −∇U (q(t)), (5.1) which can be considered as a Hamiltonian system with H : R 2N → R defined by H(z) = H(q, r) = 1 2 r 2 + U (q). An action of Γ on R 2N induced by action on R N is diagonal i.e γ(q, r) → (γq, γr). It is easy to verify that this action is symplectic, so Γ acts unitary on R 2N . Moreover, z 0 = (q 0 , r 0 ) = (q 0 ,q 0 ) = (q 0 , 0) (since we consider q 0 as a constant function) is a critical The easy block-form of the matrix J∇ 2 H(z 0 ) lets us to observe a bijective correspondence between positive eigenvalues of ∇ 2 U (q 0 ) and the pairs of purely imaginary eigenvalues of J∇ 2 H(z 0 ). In fact, if β 2 ∈ σ(∇ 2 U (q 0 )) then ±iβ ∈ σ(J∇ 2 H(z 0 )). Taking into account the above reasoning, the following theorems are consequences of Theorem 4.1.
Proof. Note, that the assumptions (A1)-(A4) are satisfied directly due to statements of the theorems above.
To complete the proofs of theorems we have to verify assumption (A5) in both cases. form Since c > 0, by Descartes rule of signs there exists exactly one positive root of this equation, say d 0 . It means that there exist one SO(2) orbit of critical points of H i.e. SO(2)(Q) where Q = (d 0 , 0, 0, 0, −ωd 0 , 0). The point Q from this orbit is chosen such that r = q 2 1 + q 2 2 = q 1 = d 0 . This orbit is obviously isolated in (∇H) −1 (0). To apply theorem 5.4 we compute the Hessian ∇ 2 H(Q). We have The Hessian is obviously degenerate (see Remark 2.4). One can see that it possesses eigenvalues 1 + ω 2 (with eigenvector [0, 1, 0, −1/ω, 0, 0] T ]) and 1 (with eigenvector [0, 0, 0, 0, 0, 1] T ). Denote be λ 1 , λ 2 , λ 3 the other three eigenvalues. If we compute the characteristic polynomial w(t) of ∇ 2 H(Q) its coefficient of the term t (which is the additive inverse of the product of eigenvalues different from the one zero-eigenvalue we have already know) equals −(1 + ω 2 ) −ω 2 V ′′ z,z (d 0 , 0) + V ′′ r,r (d 0 , 0)V ′′ z,z (d 0 , 0) − (V ′′ r,z (d 0 , 0)) 2 and substituting formulas (6.4) we obtain Hence one or three of λ i are negative. Therefore the Hessian ∇ 2 H(Q) has two or four positive eigevalues. It means that the assumption (A7.5) is satisfied. Moreover, the kernel of this Hessian is one-dimensional which provides that the orbit SO(2)(Q) is non-degenerate. Therefore the assumption (A5) is also satisfied (see the reasoning in the last paragraph of the previous section on the page 18). To summarize, all assumptions of Theorem 5.4 are satisfied. It provides the existence of periodic solutions in a nearby of an equilibrium Q in the rotating frame. These solutions correspond to a motion in a neighborhood of the geostationary orbit.
Acknowledgements. I would like to thanks prof. Sławomir Rybicki for the fruitful discussions on the topic of this article and prof. Andrzej Maciejewski for the proposition of physicalmotivated example.
The author was partially supported by the National Science Centre, Poland (Grant No. 2017/25/N/ST1/00498).