2.1 Relationship with the averaging method

Here we intend to find some relations or comparisons between the periodic solutions obtained by the continuation of the elliptic solutions (non-circular) of the Kepler problem given in theorem 2.3 with those found by using the averaging theory on Hamiltonian systems.

Initially, in order to apply the averaging method, let $h^*$ be a negative real constant such that $\mathcal {H}_0(L)=-\frac {1}{2L^2}=h^*$, so $\mathcal {H}_0^{-1}$ in a neighbourhood of $h^*$ is a diffeomorphism and $\mathcal {H}_0^{-1}(h^*)$ is a compact connected circle bundle over the base space $B(h^*)$ with projection $\pi :\mathcal {H}_0^{-1}(h^*)\rightarrow B(h^*)$ (see [Reference Yanguas, Palacián, Meyer and Dumas36] for more details). Moreover, all the solutions of the Hamiltonian system associated with the Hamiltonian (2.2) are periodic and have periods depending smoothly on $h^*$, i.e., the period is a smooth function $T=T(h^*)$.

Next, we denote by $\overline {\mathcal {H}}$ the averaging function with respect to the mean anomaly $\ell$ as

(2.26)\begin{equation} \overline{\mathcal{H}}=\frac{1}{2\pi}\int_0^{2\pi}\mathcal{H}_1(\ell,g,L,G)d\ell=\frac{1}{2\pi}\int_0^{2\pi}\mathcal{H}_1(E-e\sin E,g,L,G)L(1-e\cos E)\,{\rm d}E. \end{equation}

Thus, at the energy level $\mathcal {H}=h^*$ ($h^*<0$), the averaged system associated with (2.26) is given by

(2.27)\begin{equation} \begin{aligned} \frac{d g}{d \ell} & =\epsilon \frac{1}{({-}2h^*)^{3/2}}\frac{\partial\overline{\mathcal{H}} }{\partial G}=\epsilon \frac{1}{({-}2h^*)^{3/2}}F_1(g, G),\\ \frac{d G}{d \ell} & ={-}\epsilon \frac{1}{({-}2h^*)^{3/2}}\frac{\partial\overline{\mathcal{H}} }{\partial g}=\epsilon \frac{1}{({-}2h^*)^{3/2}}F_2(g, G). \end{aligned} \end{equation}

Next, for our problem, we have the following result as a consequence of averaging theory or Reeb's theorem for Hamiltonian systems [Reference Meyer, Palacián and Yanguas22, Reference Reeb30].

Theorem 2.6 Consider the averaged differential system (2.27) in Delaunay variables restricted to the energy level $\mathcal {H}= h$ ($h<0$). If $\bar {p}=(g_0,G_0)$ is a non-degenerate critical point, then there are smooth functions $p(\epsilon )= (\ell _0(\epsilon ), g(\epsilon ), L(\epsilon ), G(\epsilon ))$ and $T(\epsilon )$ for $\epsilon$ small with $p(0)=(\ell _0,g_0,1/\sqrt {-2h^*},G_0)$ and $T(0)=T=2\pi L_0^3$ such that the solution of (2.1) through $p(\epsilon )$ is $T(\epsilon )$-periodic.

In addition, if the characteristic exponents of the critical point $\bar {p}$ (i.e., the eigenvalues of the matrix $A=\mathbb {J}D^2\mathcal {\bar {H}}(\bar {p})$) are $\lambda _1,\lambda _2$, then the characteristic multipliers of the periodic solution through $p(\epsilon )$ are

\[ 1,1,1+\epsilon\lambda_1 T+O(\epsilon^2),1+\epsilon\lambda_2 T+O(\epsilon^2). \]

Next, we will discuss the relationship between both methods, theorem 2.3 using Poincaré continuation and theorem 2.6 using the averaging theory. Our explanation is performing under two points of view: the curve of initial conditions and the properties of the family of periodic solutions obtained by each theorem.

Without loss of generality, we will assume that the Hamiltonian (1.1) [resp. Hamiltonian (2.1)] is invariant with respect to the symmetry $S_2$, i.e., the reflection with respect to the $x$-axis. The analysis of the other symmetry $S_1$ follows using similar arguments.

Since $\ell =E-e\sin E$, then the function $\mathcal {H}_1(\ell,g,L,G)$ assumes the form $\mathcal {H}_1=\mathcal {H}_1 (E,g,L,G)$. By the invariance of the function $\mathcal {H}_1$ with respect to the symmetry $S_2$ it follows that

(2.28)\begin{equation} \mathcal{H}_1(E,g,L,G)=\mathcal{H}_1(2\pi-E,2\pi-g,L,G).\end{equation}

Differentiating both sides of the last equation with respect to $g$ and evaluating at $g=k\pi$, $k=0,1$, we obtain

(2.29)\begin{equation} \frac{\partial \mathcal{H}_1}{\partial g}(E,k\pi,L,G)={-}\frac{\partial \mathcal{H}_1 }{\partial g}(2\pi-E,k\pi,L,G) \quad k=0,1. \end{equation}

Multiplying both sides of equation (2.29) by $L^3(1-e\cos E)$ and integrating in $E$ from $0$ to $2\pi$, we obtain

(2.30)\begin{align} & \int_0^{2\pi}\frac{\partial \mathcal{H}_1 }{\partial g}(E,k\pi,L,G)L^3(1-e\cos E)\,{\rm d}E\nonumber\\ & \quad ={-}\int_0^{2\pi}\frac{\partial \mathcal{H}_1 }{\partial g}(2\pi-E,k\pi,L,G)L^3(1-e\cos E)\,{\rm d}E. \end{align}

Making the change $\bar {E }=2\pi -E$ on the right-hand side of equation (2.30), we obtain (after removing the bar)

\[ \int_0^{2\pi}\frac{\partial {\mathcal{H}}_1}{\partial g}(E,k\pi,L,G)L^3(1-e\cos E)\,{\rm d}E=0. \]

Now, from (2.26), we have

(2.31)\begin{equation} \frac{\partial\overline{\mathcal{H}} }{\partial g}|_{g=k\pi}=\int_0^{2\pi}\frac{\partial {\mathcal{H}}_1}{\partial g}(E,k\pi,L,G)L^3(1-e\cos E)\,{\rm d}E=0, \quad k=0,1. \end{equation}

Thus, $F_2(g=k\pi, G)=0$ for $k=0,1$. In addition, from equation (2.31), it follows that $\frac {\partial ^2 \overline {\mathcal {H}}}{\partial G\partial g}\Big |_{g=k\pi }=0$.

Proposition 2.7 Suppose that the Hamiltonian function $\mathcal {H}$ given in (2.1) is invariant with respect to the symmetry $S_2$. Thus,

(2.32)\begin{equation} g^{(1)}(t,{\bf Y}_0^{2,k})=L_0^3\pi F_1(g=k\pi,G_0). \end{equation}

Proof. For $\mathcal {H}_1$ given in (2.23) we have

(2.33)\begin{align} g^{(1)}(t,{\bf Y}^{2,k})\Big|_{g=k\pi}& = \int_0^{\pi}\frac{\partial {\mathcal{H}}_{1}}{\partial G}(E,g,L,G)\Big|_{g=k\pi}L^3(1-e\cos E)\,{\rm d}E\nonumber\\ & =L^3\frac{\partial }{\partial G}\left(\displaystyle\int_0^{\pi}{\mathcal{H}}_{1}(E,g,L,G)(1-e\cos E)\Big|_{g=k\pi}\,{\rm d}E\right)\nonumber\\ & =\frac{L^3}{2}\frac{\partial }{\partial G}\left(\int_0^{2\pi}{\mathcal{H}}_{1}(E,g,L,G)(1-e\cos E)\Big|_{g=k\pi}\,{\rm d}E\right)\nonumber\\ & =\frac{L^3}{2}\left(\frac{\partial }{\partial G}\int_0^{2\pi}{\mathcal{H}}_{1}(E,g,L,G)(1-e\cos E)\,{\rm d}E\right)\Big|_{g=k\pi}\nonumber\\ & ={\pi L^3}\frac{\partial \overline{H}_1 }{\partial G}\Big|_{g=k\pi}={\pi L^3}F_1(g=k\pi,G) \end{align}

Note that the equality

\begin{align*} & 2\int_0^{\pi}{\mathcal{H}}_{1}(e,E,g,L,G)(1-e\cos E)\Big|_{g=k\pi}\,{\rm d}E\nonumber\\ & \quad =\int_0^{2\pi}{\mathcal{H}}_{1}(e,E,g,L,G)(1-e\cos E)\Big|_{g=k\pi}\,{\rm d}E, \end{align*}

follows from the fact that by equation (2.28), the function $F(E)={\mathcal {H}}_{1}(e,E,g,L,G)(1-e\cos E)\Big |_{g=k\pi }$ satisfies $F(E)=F(2\pi -E)$. Finally, from (2.33) we obtain (2.32).

From the previous discussion, we can relate the $S_2$-symmetric periodic solutions obtained by theorem 2.3 with those obtained using the averaging method described in theorem 2.6. More precisely, from theorems 2.3 and 2.6 we obtain the following result.

Theorem 2.8 Consider $T=2\pi L_0^3$ and let $\varphi _{0}(t, {\bf Y}_{0}^{j,k})=(\frac {t}{L_{0}^3}, g_{0}^{j,k}, L_0, G_0)$ be a $S_j$-symmetric elliptic Keplerian solution $T$-periodic such that the conditions (a) and (b) of theorem 2.3 are satisfied.

Thus, $\bar {p}=(g_{0}^{j,k},G_0)$ is a critical point of the differential system (2.27). Furthermore, if $\bar {p}$ is an isolated critical point of (2.27) and $\frac {\partial ^2 \overline {\mathcal {H}}}{\partial g^2}(\bar {p})\neq 0$, then the elliptic Keplerian orbit $\varphi _{0}(t,{\bf Y}_0^{j,k})$ $($associated with the critical point $\bar {p})$ can be continued by Reeb's theorem to a $p(t, \bar {p}(\epsilon ))$, $T(\epsilon )$-periodic solution.

Moreover, if $1+\epsilon ^{\alpha } \lambda _1 +{\mathcal {O}}(\epsilon ^{\alpha +1}),1+\epsilon ^{\alpha }\lambda _2$ are the nontrivial characteristic multipliers of the periodic solutions $\varphi (t,{\bf Y}_{\epsilon }^{j,k};\epsilon )$ given by theorem 2.3, then the nontrivial multipliers characteristic of the periodic solutions $p(t,\bar {p}(\epsilon ))$ are

\[ 1+\epsilon^{\alpha} \lambda_1T +{\mathcal{O}}(\epsilon^{\alpha+1}),1+\epsilon^{\alpha}\lambda_2T. \]

Proof. We give the proof for the $S_2$-symmetric periodic solutions. Let $\varphi _{0}(t,{\bf Y}^{2,k})$ be an elliptic Keplerian solution near the elliptic solution $\varphi _{0}(t,{\bf Y}_0^{2,k})=(\frac {t}{L_{0}^3},0, L_0, G_0)$ such that the conditions (a) and (b) of theorem 2.3 are satisfied. By the previous discussion and by proposition 2.7 it follows that $\bar {p}=(g_0=k\pi,G_0)$ is a critical point of the average function $\bar {\mathcal {H}}$ given in (2.26).

Since $\frac {\partial ^2 \bar {\mathcal {H}}}{\partial g\partial G}\Big |_{g=k\pi }=0$, the non-degeneracy condition at the critical point $\bar {p}$ is given by

\[ \mathcal{D}=\det \mathbb{J}D^2\mathcal{H}(\bar{p})=\frac{\partial^2 \bar{\mathcal{H}}}{\partial G^2}\frac{\partial^2 \bar{\mathcal{H}}}{\partial g^2}\Big|_{g=k\pi,G=G_0}\neq 0. \]

From (2.32) it follows that

\[ \frac{\partial^2 \bar{\mathcal{H}}}{\partial G^2}(\bar{p})=\frac{1}{\pi L_0^3}\frac{\partial g^{(1)}(t,{\bf Y}_0^{2,k})}{\partial G}(\bar{p}), \]

and by condition (b) of theorem 2.3, we obtain $\frac {\partial ^2 \bar {\mathcal {H}}}{\partial G^2}(\bar {p})\neq 0$. Finally, since $\frac {\partial ^2 \overline {\mathcal {H}}}{\partial g^2}(p)\neq 0$ we obtain $\mathcal {D}\neq 0$. The conclusion about the nontrivial multipliers characteristic is immediate from theorems 2.3 and 2.6. Thus, we have concluded the proof.

We finish this section with some remarks about the similarities/differences of the periodic solutions obtained in theorem 2.3 and those given in theorem 2.8.

• • Let $\varphi _{0}(t,{\bf Y}_0^{2,k})=\left (\frac {t}{L_{0}^3},0, L_0, G_0\right )$ be an elliptic Keplerian solution and assume that the conditions (a) and (b) of theorem 2.3 are satisfied. By the previous discussion it follows that $\bar {p}=(g_0=0,G_0)$ is not necessarily a non-degenerate critical point of the system (2.26). Thus, the condition (b) of theorem 2.3 is a weaker condition when compared with the condition for the non-degeneracy of the critical point $\bar {p}$ given by theorem 2.8. For a concrete example of this situation, see remark 3.5.

• • Note that the periodic ($S_2$-symmetric) solutions obtained by theorem 2.3 item (i) have the same period $T=2\pi L_0^{3}$ that the elliptic orbit $\varphi _{0}(t,{\bf Y}_0^{2,k})$ of the unperturbed system. The solutions obtained in theorem 2.8 have period $T=2\pi L_0^{3}+\mathcal {O}(\epsilon )$.

• • The approximation of the period of the periodic solutions given in theorems 2.8 and 2.3 item (ii) can be computed as follows. If $\ell (0)=0$ and $\ell (T)=2\pi$, then $T(\varepsilon )$ must satisfy the equation

  \begin{align*} 2\pi & = \ell(T)-\ell(0)=\displaystyle\int_0^T \dot{\ell}(t)\,{\rm d}t= L_0^{{-}3} T+\varepsilon^{\alpha} L_0^{{-}3} \displaystyle\int\limits_0^{2 \pi} \dfrac{\partial \mathcal{H}_1} {\partial L}(g,L,G)\,{\rm d}\ell +O(\varepsilon^{\alpha+1})\\ & = L_0^{{-}3} T+ 2 \pi L_0^{{-}3} \varepsilon^{\alpha} \dfrac{\partial \mathcal{H}_1} {\partial L}(g,L,G)|_{L=L_0, g=g_0, G=G_0} +O(\varepsilon^{\alpha+1})\\ & = L_0^{{-}3} \left(T+ 2 \pi\varepsilon^{\alpha} \dfrac{\partial \bar{\mathcal{H}}} {\partial L}(g,L,G)|_{L=L_0, g=g_0, G=G_0}\right) +O(\varepsilon^{\alpha+1}) . \end{align*}

  By use of the implicit function theorem, it follows that $T(\varepsilon )=2\pi L_0^3+\varepsilon T^\ast +O(\varepsilon ^2)$, where $T^\ast =-2 \pi \dfrac {\partial \bar {\mathcal {H}}}{\partial L}(g,L,G)|_{L=L_0, g=g_0, G=G_0}$.

  We call attention for the initial conditions generating periodic solutions of the full system on the reduced space $\mathcal {B}(h)$ (parametrized by the coordinates $(g, G)$) obtained in theorem 2.3 and those given by theorem 2.8. The curve of initial conditions parametrized by $\epsilon$ at a fixed point $\bar {p}=(g_0^{j,k},G_0)$ as in corollary 2.8 has the form $(g(\epsilon ), G(\epsilon ))$. On the other hand, the curve obtained in theorem 2.3 at the same point $\bar {p}$ has the form $(k \pi, \overline {G}(\epsilon ))$. In figure 3, we illustrate these two families of initial conditions in a neighbourhood of a fixed critical point $\bar {p}$ on the reduced space.

• • The periodic solutions given by theorem 2.8 are not necessarily symmetric. The authors in reference [Reference Yanguas, Palacián, Meyer and Dumas36] (see § 2.4) did the inverse process of this section, i.e., they showed that the Keplerian solutions associated with the critical point of the averaging system can be continued to a symmetric periodic solution under some restrictions. In fact, the technique of these authors consists in the use of the symmetries of the problem combined with averaging, reduction theory and the use of the implicit function theorem of Arenstorf. But, the use of the theorem of Arenstorf requires a suitable choice of the parameter $\epsilon$. In fact, the value of the parameter $\epsilon$ must be chosen in a discrete set. Thus, the symmetric periodic solutions analysed in [Reference Yanguas, Palacián, Meyer and Dumas36] have the following properties: the existence of the symmetric solutions is conditioned only for some values of the parameter $\epsilon$; the period of these symmetric periodic solutions is very large and is given by $\tau (\epsilon )=\beta T+\mathcal {O}(\epsilon )$ (here, the parameter $\beta$ is a positive and very large integer and $T$ is the period of the Keplerian orbit).

  Figure 3.

  Illustration of the projection curves of initial conditions in a neighbourhood of the point $\bar {p}=(g_0=\pi, G_0)$ parametrized by the coordinates $(g, G)$. The blue curve $(g(\epsilon ), G(\epsilon ))$ represents the curve of initial conditions obtained by theorem 2.8. The red curve $( \pi, \overline {G}(\epsilon ))$ is the projection of the curve of initial conditions given by theorem 2.3 over the reduced space and it is associated with the $S_2$-symmetric periodic solutions.

  

  The last paragraph shows that there are important differences between the symmetric solutions obtained in [Reference Yanguas, Palacián, Meyer and Dumas36] and those obtained in § 2.4, since these can be obtained for all values of $\epsilon$ (sufficiently small) and there is no restriction on the period $T$.

3.1 Planar hydrogen atom with Stark and quadratic Zeeman effect

We consider a simple atomic system, namely hydrogen atom interacting with time-independent and external fields that include electric and magnetic fields. In this formulation, the magnetic field is usually applied perpendicular to the $xy$-plane, while the static electric field is applied along the $x$-axis. See [Reference De Bustos, Guirao, Vera and Vigo-Aguilar11, Reference Ganesan and Gȩbarowski12, Reference Gusev, Samoilov, Rostovtsev and Vinitsky15] or [Reference Palacián24] for more details on the formulation of this problem. We call this problem simply by HCEM problem, whose Hamiltonian has the form

(3.1)\begin{equation} H(x,y,p_x,p_y)=\frac{1}{2}(p_{x}^2+p_{y}^2)-\frac{1}{\sqrt{x^2+y^2}}+\frac{\tilde{B}}{2}(yp_x-xp_y)+\frac{B^2}{8}(x^2+y^2)-F x. \end{equation}

This Hamiltonian function depends on two parameters $\tilde {B}$ and $F$. The term $Fx$ is the electrostatic potential describing the Stark effect while the other terms having the parameter $\tilde {B}$ refer to the linear and quadratic Zeeman effect (see [Reference Ganesan and Gȩbarowski12, Reference Palacián24] for more details for definition of the physical constants). An important point in this problem is that the angular momentum is not a conserved quantity. We pint out that the Hamiltonian function (3.1) is invariant under (only) the symplectic reflection $S_2:(x,y,p_x,p_y)\rightarrow (x,-y,-p_x,p_y)$. We will consider the hydrogen atom in crossed electric and magnetic field problem for weak magnetic and electric fields. This is achieved scaling $\tilde {B}$ and $F$ in the following way:

\[ \tilde{B}=2 B \epsilon, \quad F=\gamma \epsilon, \]

where $\epsilon$ is a parameter sufficiently small. Considering this scaling, the Hamiltonian function (3.1) assumes the form

(3.2)\begin{equation} H(x,y,p_x,p_y)=\displaystyle\frac{1}{2}(p_{x}^2+p_{y}^2)-\frac{1}{\sqrt{x^2+y^2}}+\epsilon\left[B(yp_x-xp_y)-\gamma x\right]+\epsilon^2\frac{B}{2}(x^2+y^2). \end{equation}

To show the existence of symmetric periodic solutions, first as in the previous case, we write the Hamiltonian function (3.2) in mixed variables involving polar and Delaunay variables, so obtaining the Hamiltonian function

\[ \mathcal{H}={-}\frac{1}{2L^2}+\epsilon\mathcal{H}_1(E,g,L,G)+\mathcal{O}(\epsilon ^2), \]

where

(3.3)\begin{align} \mathcal{H}_1(E,g,L,G)& ={-}B G - r \gamma \cos(f+g),\nonumber\\ & ={-}B G+a \gamma \left(\sqrt{1-e^2} \sin E \sin (g)+\cos (g) (e-\cos E)\right), \end{align}

$r=a(1-e^2)/(1+e\cos f)$ and $e=\sqrt {1-(G^2/L^2)}$. The main result about the existence of $S_2$-symmetric periodic solutions for the $HCEM$ problem (3.2) is the following.

Proof. Maintaining the notation of § 2, first we use expression (2.24) for calculations involving the derivative $\frac {\partial {\mathcal {H}}_{1}}{\partial G}$. After some calculations, we obtain

(3.4)\begin{equation} \frac{\partial {\mathcal{H}}_{1}}{\partial G}={-}B+G \gamma \cos g \left(-\frac{\sin ^2E}{e-\text{ee}^2 \cos E}-\frac{1}{e}\right)-\frac{G \gamma \sin E \sin g (e-\cos E)}{e \sqrt{1-e^2} (e \cos E-1)}. \end{equation}

Next, we evaluate it in the solution $\varphi _{0}(\tau, {\bf Y}^{2,k})$, and after some simplifications, we arrive at

(3.5)\begin{align} & \frac{\partial {\mathcal{H}}_{1}}{\partial G}(\varphi_{0}(\tau, {\bf Y}^{2,k})) \nonumber\\ & \quad =\frac{B e+({-}1)^kG \gamma-e \cos (E) (B e+({-}1)^{k+1}G \gamma )+({-}1)^kG \gamma \sin ^2(E) }{e (e \cos (E)-1)},\end{align}

where $e=\sqrt {1-G^2/L^2}$, $G=G_0+\delta G$, $L=L_0+\delta L$. For a fixed value of $0< e<1$ the function given by (3.5) is continuous and differentiable with respect to $E$ and then the integral of (3.5) can be calculated explicitly. For the integration of equation (3.5), we use the change of variables (2.25). After the integration of equation (3.5) and some simplifications, we arrive at

(3.6)\begin{align} g^{(1)}(T/2,{\bf Y}_0^{2,k})& = \displaystyle\int_{0}^{\pi}\frac{\partial {\mathcal{H}}_{1}}{\partial G}(\varphi_{0}(\tau, {\bf Y}^{2,k}))|_{{\bf Y}^{2,k}={\bf Y}_{0}^{2,k}}L_{0}^{3}(1-e_0\cos E)\,{\rm d}E\nonumber\\ & =\pi L_0^3 \left(- B+({-}1)^{k+1}\frac{3 \gamma G_0 L_0}{2\sqrt{L_0^2-G_0^2}}\right)\nonumber\\ & =\pi L_0^3 \gamma \left(- \frac{B}{\gamma}+({-}1)^{k+1}\frac{3 G_0 L_0}{2\sqrt{L_0^2-G_0^2}}\right). \end{align}

To find the solutions (in the variable $G_0$) of $g^{(1)}(T/2,{\bf Y}_0^{2,k})=0$, we define the normalized parameter $B_1=-B/\gamma$. Moreover, we introduce the auxiliary functions

(3.7)\begin{equation} f_1^k(G_0)=({-}1)^{k+1}\frac{3 G_0 L_0}{2\sqrt{L_0^2-G_0^2}},\quad f_2(G_0)=B_1,\quad k=0,1, \end{equation}

defined in $(-L_0,L_0)$. Of course, $g^{(1)}(T/2,{\bf Y}_0^{2,k})=0$, if and only if,

(3.8)\begin{equation} f_1^k(G_0)={-}f_2(G_0), \end{equation}

for some $G_0\in (-L_0,L_0)$. Due to straightforward properties of the function $f_1^k$, we obtain the following possibilities for the solutions of (3.8):

• • If $k=0$ and $B_1<0$ (resp. $B_1>0$), then there is a unique solution $G_0^{(0)}=\frac {2 B_1 L_0}{\sqrt {4 B_1^2+9 L_0^2}}<0$ (resp. $G_0^{(0)}=\frac {2 B_1 L_0}{\sqrt {4 B_1^2+9 L_0^2}}>0$).

• • If $k=1$ and $B_1<0$ (resp. $B_1>0$), then there is a unique solution $G_0^{(1)}=-\frac {2 B_1 L_0}{\sqrt {4 B_1^2+9 L_0^2}}>0$ (resp. $G_0^{(1)}=-\frac {2 B_1 L_0}{\sqrt {4 B_1^2+9 L_0^2}}<0$).

Observe that $0<|G_0^{(0)}|=|G_0^{(1)}|< L_0$, whenever $B_1\neq 0$ and $\gamma \neq 0$. Thus, we have verified the condition (a) of theorem 2.3.

Now, we are going to verify the condition (b) given by theorem 2.3. Consider the equation (2.24) and the chain rule to differentiate (3.5) with respect to $\delta G$. After that, we evaluate $\displaystyle \frac {\partial }{\partial \delta G}\left (\frac {\partial \mathcal {H}_1}{\partial G}\right )$ on the solution $\varphi _{0}(\tau, {\bf Y}_0^{2,k}))$. Next, using the change (2.25), after integration and simplification, we arrive at

(3.9)\begin{align} \frac{\partial g^{(1)}}{\partial \delta G}\Big |_{{\bf Y}^{2,k}={\bf Y}_{0}^{2,k}} & =\displaystyle\int_{0}^{\pi}\frac{\partial }{\partial \delta G}\left(\frac{\partial \mathcal{H}_1}{\partial G}(\varphi_{0}(\tau, {\bf Y}^{2,k}))\right)\Big |_{{\bf Y}^{2,k}= {\bf Y}_{0}^{2,k}}L_0^3(1-e_0\cos E)\,{\rm d}E \nonumber\\ & ={-}\frac{3 \pi \gamma L_0^6}{2 \left(L_0^2-G_0^2\right)^{3/2}}. \end{align}

Of course, for all $L_0,\gamma \neq 0$ we have $\frac {\partial g^{(1)}}{\partial \delta G}\Big |_{{\bf Y}_{0}^{2,k}}\neq 0$ and the item (b) of theorem 2.3 is verified. Thus, we obtain the existence of two $1$-parameter (on $\epsilon$) families of initial conditions

\[ {\bf Y}_{\epsilon}^{2,k}=\left(0, k\pi, L_0+\delta L^{(k)}(\epsilon), G_0^{(k)}+\delta G^{(k)}(\epsilon)\right), \quad k=0,1, \]

where $G_0^{(k)}=(-1)^{k}\frac {2 B_1 L_0}{\sqrt {4 B_1^2+9 L_0^2}}$, such that each of them gives us a $S_2$-symmetric of second-kind periodic solution with period $T=2\pi L_0^3$. In addition, we have two $2$-parameters (on $\epsilon$ and $\delta L$) families of initial conditions

\[ {\bf Y}_{\epsilon,\delta L}^{2,k}=(0, k\pi, L_0+\delta L^{(k)},G_0^{(k)}+\delta G^{(k)}(\epsilon,\delta L^{(k)})), \quad k=0,1, \]

such that, each of them gives us $S_2$-symmetric periodic solution of second kind with period $\overline {T}$ close to $T=2\pi L_0^3$.

Now, we discuss the stability of the previous periodic solutions. After some algebraic manipulation, the averaging function $\overline {H}$ given in (2.10) here has the form

(3.10)\begin{equation} \overline{H}(X_1,X_2)=\pi L_0^3 \left(3 \gamma L_0 \cos (g_0^{2,k}+X_1) \sqrt{L_0^2-(G_0+X_2)^2}+2 B (G_0+X_2)\right). \end{equation}

By the definition of the matrix $A$ in (3.10), we obtain

(3.11)\begin{equation} A=\left(\begin{array}{cc} 0 & -({-}1)^k\dfrac{3 \pi \gamma L_0^6}{\left(L_0^2-G_0^2\right)^{3/2}} \\ ({-}1)^k 3 \pi \gamma L_0^4 \sqrt{L_0^2-G_0^2} & 0 \\ \end{array}\right). \end{equation}

Thus, the corresponding characteristic multipliers of the prograde and retrograde $S_2$-symmetric periodic solutions are

\[ 1,1, 1+\epsilon \frac{3 i \pi \gamma L_0^5}{\sqrt{L_0^2-G_0^2}}+\mathcal{O}(\epsilon^2),1-\epsilon \frac{3 i \pi \gamma L_0^5}{\sqrt{L_0^2-G_0^2}}+ \mathcal{O}(\epsilon^2), \]

where $G_0=G_0^{(k)}$. Thus, in any case the $S_2$-symmetric periodic solutions are linearly stable.

3.2 Anisotropic two-body problem under Seeliger's potential

The gravitational potential of the two-body problem due to Seeliger's theory is given by

(3.12)\begin{equation} V({\bf q})={-}\frac{A}{\|{\bf q}\|}\,{\rm e}^{{-}K\|{\bf q}\|}, \end{equation}

where ${\bf q}$ is the vector between the two masses $m_1$ and $m_2$, $A$ and $K$ are positive constants. See [Reference Mioc and Rusu23, Reference Paşca and Valls26, Reference Popescu, Paşca, Mioc and Popescu29] for more details on the formulation of this problem. We mention that a field featured by a (3.12)-like potential has larger physical implications. London's theory of superconductivity involves an electromagnetic potential of this form. Debye–Huckel's theory of screening in electrolytes leads to a similar screened potential. See [Reference Paşca and Valls26] and references therein for more physical implications of this problem.

In this paper, following the formulation given in [Reference Paşca and Valls26], we consider the Seeliger potential in an anisotropic space, where the potential function (3.12) depends on the parameter $\mu$ that measures the strength of anisotropy. More precisely, we consider the Hamiltonian

(3.13)\begin{equation} H(x,y,p_x,p_y)=\frac{1}{2}(p_{x}^2+p_{y}^2)-\frac{A}{\sqrt{\mu x^2+y^2}}e^{{-}K\sqrt{\mu x^2+y^2}}. \end{equation}

We call this problem as anisotropic Seeliger's Hamiltonian or shortly ASH. It is clear that the Hamiltonian function (3.13) is invariant under the symmetries $S_1$ and $S_2$.

To obtain a convenient approach of the Hamiltonian (3.13), we introduce the scaling $\mu =1-\mu _0\epsilon ^2$ and $K=\epsilon \xi$ for $\epsilon$ small. In addition, we make the change ${\bf {p}}=\sqrt [3]{A}{\bf {p}}$, ${\bf {q}}=\sqrt [3]{A}{\bf {q}}$, that is, a $1/A^{2/3}-$symplectic change. After this scaling, we develop the resulting Hamiltonian in a Taylor series in $\epsilon$ around $\epsilon =0$. Thus, eliminating the constant terms and setting $\kappa =\xi A^{1/3}$, ending up with

(3.14)\begin{align} H(x,y,p_x,p_y)& =\frac{1}{2}(p_{x}^2+p_{y}^2)-\frac{1}{\sqrt{x^2+y^2}}\nonumber\\ & \quad - \epsilon ^2 \left(\frac{1}{2} \kappa^2 \sqrt{x^2+y^2}+\frac{x^2\mu_0}{2 \left(x^2+y^2\right)^{3/2}}\right)+\mathcal{O}(\epsilon^3). \end{align}

In the mixed polar and Delaunay coordinates, the Hamiltonian (3.14) assumes the form

(3.15)\begin{equation} \mathcal{H}={-}\frac{1}{2L^2}+\epsilon^2 \mathcal{H}_1(r,\varphi,g,L,G)+\mathcal{O}(\epsilon ^3), \end{equation}

where

(3.16)\begin{equation} {\mathcal{H}}_1(r,f,L,G)={-}\frac{1}{2}\kappa^2 r-\frac{\cos^2(f+g)}{2r}, \end{equation}

with as customary $r=a(1-e^2)/(1+e\cos f)$ and $e=\sqrt {1-(G^2/L^2)}$.

Applying theorem 2.3, we obtain the following result about the existence of symmetric periodic solutions of second kind for the anisotropic Seeliger problem (3.14).

Proof. We continue maintaining the notation of theorem 2.3. First, we eliminate the dependence of $\mathcal {H}_1$ given in (3.16) in the variables $r$ and $f$ using the auxiliary expressions (2.22). We infer that

(3.17)\begin{align} {\mathcal{H}}_{1}(E,g,L,G)& =\frac{1}{8 a (e \cos E-1)^3}\left[\vphantom{\left. -2 e \sqrt{1-e^2} \mu_0 \cos (\text{Ec}+2 g)\right]}2 (e \cos E-1)^2 \left(a^2 \left(e^2+2\right) \kappa ^2\right.\right.\nonumber\\ & \quad \left.\left.+\,a^2 e \kappa ^2 (e \cos (2 E)-4 \cos E)+\mu_0\right)\right.\nonumber\\ & \quad+2 e \sqrt{1-e^2} \mu_0\cos (E-2 g)-\sqrt{1-e^2} \mu_0\cos (2 (E-g))\nonumber\\ & \quad +\sqrt{1-e^2} \mu_0 \cos (2 (E+g))\nonumber\\ & \quad\left.+\,\mu_0 \cos (2 g) \left(-\left(e^2-2\right) \cos (2 E)+3 e^2-4 e \cos E\right)\right.\nonumber\\ & \quad\left. -\,2 e \sqrt{1-e^2} \mu_0 \cos (\text{Ec}+2 g)\right]. \end{align}

Next, for the calculations involving the derivative $\frac {\partial {\mathcal {H}}_{1}}{\partial G}$, we use expression (2.24) and the chain rule. We obtain

(3.18)\begin{align} \displaystyle\frac{\partial {\mathcal{H}}_{1}}{\partial G}& =\displaystyle \frac{G}{4 a^2 e (e \cos E-1)^5}\left[\frac{3}{2}\cos E(e \cos E-1)\left(2(e \cos E-1)^2\left(a^2 \left(e^2+2\right) \kappa ^2+\mu_0+a^2 e \kappa ^2\right.\right.\right.\nonumber\\ & \quad \times e \cos (2 E)-4 \cos E) +\mu_0 \cos (2 g) \left(-\left(e^2-2\right) \cos (2 E)+3 e^2-4 e \cos E\right)\nonumber\\ & \quad\left.+\, \mu_0 \sqrt{1-e^2}\left[ e \cos (E-2 g)-\cos (2 (E-g))+ \cos (2 (E+g))-2 e \cos (E+2 g)\right]\right)\nonumber\\ & \quad+\sin E\left(\frac{1}{2}\left({-}2 e (e \cos E-1)^2 \left(a^2 \left(e^2+2\right) \kappa ^2+a^2 e \kappa ^2 (e \cos (2 E)-4 \cos E)-\mu_0\right)\right.\right.\nonumber\\ & \quad+\left.\,\mu_0 \cos (2 g) \left(\left(8-12 e^2\right) \cos E+e \left(\left(2 e^2-1\right) \cos (2 E)+8 e^2-5\right)\right)\right)\sin E+3 e \left(e^2-1\right)\mu_0\nonumber\\ & \quad\times\sqrt{1-e^2} \mu_0\cos (E) \sin (2 g) \left(2 \left(e^2+1\right) \cos E-e (\cos (2 E)+3)\right)+\sin ^3(E) \cos (2 g)\nonumber\\ & \quad\left.-\,2 \sqrt{1-e^2} \mu_0 \sin ^2(E) \sin (2 g) \left({-}3 e^2+2 e \cos E+1\right)\right)-(e \cos E-1)^2\nonumber\\ & \quad\times \left(8 a^2 e^3 \kappa ^2 \cos ^4E-24 a^2 e^2 \kappa ^2 \cos ^3E+2 e \cos ^2E \left(12 a^2 \kappa ^2+\mu_0\right)\right.\nonumber\\ & \quad \left. -e\mu_0(\cos (2E)-3) \cos (2 g)\right.\nonumber\\ & \quad+\,\left.\left.2 \cos E \left({-}4 a^2 \kappa ^2+\mu_0 \left(\frac{e \sin E \sin (2 g)}{\sqrt{1-e^2}}+\sin ^2(g)-\cos ^2(g)\right)-\mu_0\right)\right.\right.\nonumber\\ & \quad\left.\left. +\frac{2 \left(1-2 e^2\right) \mu_0 \sin E \sin (2 g)}{\sqrt{1-e^2}}\right)\right]. \end{align}

Next, we evaluate (3.18) on the solution $\varphi _{0}(\tau, {\bf Y}^{j,k})$. After some simplifications, we arrive at

(3.19)\begin{align} \frac{\partial {\mathcal{H}}_{1}}{\partial G}(\varphi_{0}(\tau, {\bf Y}^{1,k}))& = \displaystyle\frac{G_0 ({-}e + \cos E)}{16 a^2 e ( e \cos E-1)^5}\left[ 4 e \cos E \left(\mu_0-2 a^2 \left(3 e^2+4\right) \kappa ^2\right)\right.\nonumber\\ & \quad \left. +\,a^2 \left(3 \left(e^2+8\right) e^2+8\right) \kappa ^2\right.\nonumber\\ & \quad +4 (3 e^2-5) \mu_0+4 \cos (2 E) \left(e^2 \left(a^2 \left(e^2+6\right) \kappa ^2-3 \mu_0\right)+5 \mu_0\right)\nonumber\\ & \quad\left.-\,4 e \cos (3 E) \left(2 a^2 e^2 \kappa ^2+\mu_0\right)+a^2 e^4 \kappa ^2 \cos (4E)\right], \end{align}

(3.20)\begin{align} \frac{\partial {\mathcal{H}}_{1}}{\partial G}(\varphi_{0}(\tau, {\bf Y}^{2,k}))& = \displaystyle\frac{G_0 ({-}e + \cos E)}{16 a^2 e (e \cos E-1)^5}\left[4 e \cos E \left(3\mu_0-2 a^2 \left(3 e^2+4\right) \kappa ^2\right)\right.\nonumber\\ & \quad \left. +\,a^2 \left(3 \left(e^2+8\right) e^2+8\right) \kappa ^2\right.\nonumber\\ & \quad+ 4 \left(3-4 e^2\right) \mu_0+4 \cos (2 E) \left(e^2 \left(a^2 \left(e^2+6\right) \kappa ^2+2 \mu_0\right)-5 \mu_0\right)\nonumber\\ & \quad\left.-\,4 e \cos (3 E) \left(2 a^2 e^2 \kappa ^2-\mu_0\right)+a^2 e^4 \kappa ^2 \cos (4 E)\right], \end{align}

where $a^2=L$, $G=G_0+\delta G$ and $L=L_0+\delta L$. Note that for $0< e<1$ the functions in (3.19)–(3.20) are continuous and differentiable with respect to the variable $E$ and it can be integrated on $E$. Next, for the integration of the previous two equations, we use the change (2.25). Therefore, after integration of (3.19)–(3.20) in the new variable $E$ and some simplifications, we get

(3.21)\begin{align} g^{(1)}(T/2,{\bf Y}_0^{j,k})& = \int_{0}^{\pi}\frac{\partial {\mathcal{H}}_{1}}{\partial G}(\varphi_{0}(\tau, {\bf Y}^{j,k}))|_{{\bf Y}^{j,k}={\bf Y}_{0}^{j,k}}L_{0}^{3}(1-e_0\cos E)\,{\rm d}E\nonumber\\ & =\frac{1}{2} \pi G_0 L_0^3 \left(\kappa ^2-({-}1)^{j}\frac{\left(e_0^2+2 \sqrt{1-e_0^2}-2\right) \mu_0}{a^2 e_0^4 \sqrt{1-e_0^2}}\right). \end{align}

Replacing $e_0=\sqrt {L_0^2-G_0^2}/L_0$ on equation (3.21), we obtain

(3.22)\begin{align} & g^{(1)}(T/2,{\bf Y}_0^{j,k})\nonumber\\ & \quad =\frac{\pi G_0 L_0^2 \left(G_0^4 |G_0|\kappa ^2 L_0 -2 G_0^3 \kappa ^2 L_0^3+|G_0| \kappa ^2 L_0^5-({-}1)^j\mu_0( G_0^2+L_0^2-2L_0|G_0|)\right)}{2 |G_0| (G_0-L_0)^2 (G_0+L_0)^2}. \end{align}

Now, we must search the solutions $G_0$ of the equation $g^{(1)}(T/2,{\bf Y}_0^{j,k})=0$, separately, in the two domains

(3.23)\begin{equation} D_1=\{(L_0,G_0) \in \mathbb{R}^2; 0< G_0< L_0\}, \quad D_2=\{(L_0,G_0) \in \mathbb{R}^2\in \mathbb{R}; -L_0< G_0<0\}. \end{equation}

First, in the domain $D_1$( $G_0>0$) we need to solve the equation

(3.24)\begin{equation} g^{(1)}(T/2,{\bf Y}_0^{j,k})= \frac{\pi L_0^2 \left[L_0 \kappa^2 G_0^3+2L_0^2\kappa^2G_0^2+L_0^3\kappa ^2G_0+({-}1)^j\mu_0\right]}{2 (G_0+L_0)^2}=0. \end{equation}

The solutions of equation (3.24) are the roots of the polynomial equation

(3.25)\begin{equation} p_j(G_0)= L_0 \kappa^2 G_0^3+2L_0^2\kappa^2G^2+L_0^3\kappa ^2G_0+({-}1)^j\mu_0, \end{equation}

in the interval $I_p=[0,L_0]$. Initially, we consider the case $j=1$. If $\mu _0<0$, then equation (3.25) has no solution. A simple inspection shows that

\[ p_1'(G_0)=L_0\kappa^2(3 G_0^2+4 L_0 G_0+ L_0^2) >0, \text{ for all } G_0\in I_p. \]

Moreover, if $\mu _0>0$ and $L_0 > \mu _0^{1/4}/\sqrt {2\kappa }$, then $p_1(0)=-\mu _0<0$ and $p_1(L_0)=-\mu _0 + 4 \kappa ^2 L_0^4>0$. Thus, for $j=1$, $\mu _0>0$ and $L_0 > \mu _0^{1/4}/\sqrt {2\kappa }$, there exists only one solution $G_{0,1}=G_{0,1}(\kappa,L_0,\mu _0)\in (0,L_0)$ of the polynomial equation $p_1(G_0)$.

On the other, for the case $j=2$, $\mu _0<0$ and $L_0 > (-\mu _0)^{1/4}/\sqrt {2\kappa }$, there exists only one solution $G_{0,2}=G_{0,2}(\kappa,L_0,\mu _0)\in (0,L_0)$ of the polynomial equation $p_2(G_0)$.

Next, we verify the condition of non-degeneracy (b) of theorem 2.3. Again, we use the auxiliary expression (2.22) and the chain rule for the calculus involving the derivative of equations (3.19)–(3.20) with respect to $\delta G$. After that, we evaluate $\frac {\partial }{\partial \delta G}\left (\frac {\partial \mathcal {H}_1}{\partial G}\right )$ on the solution $\varphi _{0}(t, {\bf Y}_{0}^{j,k}))$. Making the change of variable $t=L_{0}^{3}(E-e_0\sin E)$, after the integration $\frac {\partial ^2 \mathcal {H}_1}{\partial G\partial \delta G}(\varphi _{0}(t, {\bf Y}_0^{j,k}))$ and some simplification, we arrive to

(3.26)\begin{align} & \frac{\partial g^{(1)}}{\partial \delta G}(\varphi_{0}(\tau, {\bf Y}^{j,k}))\Big|_{{\bf Y}^{j,k}={\bf Y}_{0}^{j,k}}\nonumber\\ & \quad =\int_{0}^{\pi}\frac{\partial }{\partial \delta G}\left(\frac{\partial \mathcal{H}_1}{\partial G}(\varphi_{0}(\tau, {\bf Y}^{j,k}))\right)\Big |_{{\bf Y}^{j,k}={\bf Y}_{0}^{j,k}}L_0^3(1-e_0\cos E)\,{\rm d}E\nonumber\\ & =\pi \frac{L_0^3}{2 a^3 e_0^6} \left(a^3 e_0^6 \kappa ^2+({-}1)^{j+1}\frac{a \left(e_0^2+2 \sqrt{1-e_0^2}-2\right) e_0^2 \mu_0}{\sqrt{1-e_0^2}}({-}1)^{j+1}\right.\nonumber\\ & \quad\left. \frac{ \left(3 e_0^4+4 \left(2 \sqrt{1-e_0^2}-3\right) e_0^2-8 \sqrt{1-e_0^2}+8\right) G_0^2\mu_0}{\left(1-e_0^2\right)^{3/2}}\right) \end{align}

Replacing $a=L_0^2$ and $e_0=\sqrt {L_0^2-G_0^2}/L_0$ on equation (3.26), we obtain

(3.27)\begin{align} & \frac{\partial g^{(1)}}{\partial \delta G}(\varphi_{0}(\tau, {\bf Y}^{j,k}))\Big |_{{\bf Y}^{j,k}={\bf Y}_{0}^{j,k}}\nonumber\\ & \quad =\int_{0}^{\pi}\frac{\partial }{\partial \delta G}\left(\frac{\partial \mathcal{H}_1}{\partial G}(\varphi_{0}(\tau, {\bf Y}^{j,k}))\right)\Big |_{{\bf Y}^{j,k}={\bf Y}_{0}^{j,k}}L_0^3(1-e_0\cos E)\,{\rm d}E\nonumber\\ & =\frac{\pi L_0^3}{2 \left(L_0^2-G_0^2\right)^3} \left[({-}1)^{j+1}\frac{\mu_0 \left(G_0^2-L_0^2\right) \left(G_0^2+L_0^2-2 L_0 | G_0|\right)}{|G_0| L_0}\right.\nonumber\\ & \left.\quad +\,({-}1)^j\frac{ \mu_0 \left(6 G_0^2 L_0^2-8 G_0^2 L_0 | G_0| +3 G_0^4-L_0^4\right)}{|G_0| L_0}\right.\nonumber\\ & \quad\left.-\,\kappa ^2 \left(G_0^2-L_0^2\right)^3\right] \end{align}

Considering $G_0>0$ in (3.27) we infer that

(3.28)\begin{equation} \frac{\partial g^{(1)}}{\partial \delta G}(\varphi_{0}(\tau, {\bf Y}^{j,k}))\Big |_{{\bf Y}^{j,k}={\bf Y}_{0}^{j,k}}=\displaystyle\frac{\pi L_0^2 \left[({-}1)^{j-1}2\mu_0+\kappa^2 L_0 (G_0+L_0)^3\right]}{2(G_0+L_0)^3}. \end{equation}

Now, if $\mu _0\!>\!0$ (resp. $\mu _0\!<\!0)$, then $\frac {\partial g^{(1)}}{\partial \delta G}(\varphi _{0}(\tau, {\bf Y}_0^{1,k}))\!>\!0$ (resp. $\frac {\partial g^{(1)}}{\partial \delta G}(\varphi _{0}(\tau, {\bf Y}_0^{2,k}))\!>\!0$) for all $G_0\in (0,L_0)$. Therefore, by theorem 2.3, we obtain the existence of a $1$-parameter (on $\epsilon$) family (and a $2$-parameter on $\epsilon$ and $\delta L$) of initial conditions [as in (2.15) and (2.16)] such that each of them gives rise to $S_j$-symmetric (prograde) periodic solution.

Second, for the study of the retrograde solutions ($G_0<0$), for each $j=1,2$, we need to search solutions of (3.22) in the domain $D_2$. Equation (3.22) for $G_0<0$ assumes the form

(3.29)\begin{equation} g^{(1)}(T/2,{\bf Y}_0^{j,k})= \frac{\pi L_0^2 \left[L_0 \kappa^2 G_0^3-2L_0^2\kappa^2G_0^2+L_0^3\kappa ^2G_0+({-}1)^{j-1}\mu_0\right]}{2 (G_0-L_0)^2}. \end{equation}

The negative solutions on the variable $G_0$ of $g^{(1)}(T/2,{\bf Y}_0^{j,k})=0$ are the solutions of the polynomial equation

(3.30)\begin{equation} q_j(G_0)=L_0 \kappa^2 G_0^3-2L_0^2\kappa^2G^2+L_0^3\kappa ^2G_0+({-}1)^{j-1}\mu_0=0. \end{equation}

It is easy to check that $q_j(G_0)=-p_j(-G_0)$, $-L_0< G_0<0$ and $j=1,2$. Thus, for each $j=1,2$, there exists a unique negative solution

\[ \overline{G}_{0,j}=\overline{G}_{0,j}(\kappa,L_0,\mu_0)={-}G_{0,j}(\kappa,L_0,\mu_0), \]

defined in the interval $(-L_0,0)$, since for $j=1$, $L_0 > \mu _0^{1/4}/\sqrt {2\kappa }$ and for $j=2$, $L_0 > (-\mu _0)^{1/4}/\sqrt {2\kappa }$.

Now, we analyse the condition (b) given in theorem 2.3. To calculate $\frac {\partial }{\partial \delta G}\left (\frac {\partial \mathcal {H}_1}{\partial G}\right )(\varphi _{0}(t, {\bf Y}_{0}^{j,k}))$, we proceed as in the previous case and after some manipulation we get

(3.31)\begin{align} \frac{\partial g^{(1)}}{\partial \delta G}\Big |_{{\bf Y}^{j,k}={\bf Y}_{0}^{j,k}}& = \displaystyle\int_{0}^{\pi}\frac{\partial }{\partial \delta G}\left(\frac{\partial \mathcal{H}_1}{\partial G}\right)(\varphi_{0}(t, {\bf Y}^{j,k}))\Big |_{{\bf Y}^{j,k}={\bf Y}_{0}^{j,k}}L_0^3(1-e_0\cos E)\,{\rm d}E\nonumber\\ & = \frac{\pi L_0^2 \left[({-}1)^{j-1}2\mu_0+\kappa^2 L_0 (L_0-{G_0})^3\right]}{2(L_0-G_0)^3}. \end{align}

If $\mu _0>0$ (resp. $\mu _0<0)$, then $\frac {\partial g^{(1)}}{\partial \delta G}(\varphi _{0}(t, {\bf Y}_0^{1,k}))>0$ (resp. $\frac {\partial g^{(1)}}{\partial \delta G}(\varphi _{0}(t, {\bf Y}_0^{2,k}))>0$) for all $G_0\in (-L_0,0)$. Thus, the conditions (a) and (b) of theorem 2.3 are satisfied and it is guaranteed of the existence of a $1$-parameter (on $\epsilon$) (resp. a $2$-parameter on $\epsilon$ and $\delta L$) family of initial conditions such that each of them gives us $S_j$-symmetric retrograde periodic solution of the Hamiltonian (3.14).

Now, we analyse the type of stability of the previous symmetric periodic solutions. After some algebraic manipulation, the averaging function $\overline {H}$ given in (2.10) for the Hamiltonian problem (3.14), in Delaunay variables, has the form

(3.32)\begin{align} \overline{H}(X_1,X_2)& =\frac{-\pi}{2} L_0 \left[\mu_0\frac{\cos (2 (g_0^{j,k}+X_1)) \left(( G_0+X_2)^2-2L_0 | G_0+X_2| +L_0^2\right)}{L_0^2-( G_0+X_2)^2}\right.\nonumber\\ & \left.\quad+\,\kappa ^2L_0^2 \left(3 L_0^2-( G_0+X_2)^2\right)+\mu_0\vphantom{\left[\mu_0\frac{\cos (2 (g_0^{j,k}+X_1)) \left(( G_0+X_2)^2-2L_0 | G_0+X_2| +L_0^2\right)}{L_0^2-( G_0+X_2)^2}\right.}\right]. \end{align}

First, we consider equation (3.32) for prograde solutions ($G_0>0$). Therefore, the matrix $A$ defined in (2.11) takes the form

(3.33)\begin{align} A_P^{(j)} & = \mathbb{J}\left( \frac{\partial^2 \bar{H}}{\partial X_k \partial X_l} \right)_{X=0}\nonumber\\ & \quad =\left( \begin{array}{cc} 0 & \displaystyle L_0^2\pi\left(L_0\kappa^2+\dfrac{2({-}1)^{j-1}\mu_0}{(G_0+L_0)^3}\right) \\ \dfrac{2 \pi L_0 ({-}1)^{j-1}\mu_0 (L_0-G_0)}{G_0+ L_0} & 0 \\ \end{array}\right), \end{align}

for $j=1,2$, whose eigenvalues are

\[ \lambda_{1,2}^{(j)}={\pm} \frac{ \sqrt{2} \pi L_0^{3/2} \sqrt{({-}1)^{j-1}\mu_0(L_0-G_0)} \sqrt{\kappa^2 L_0 (G_0+L_0)^3+2({-}1)^{j-1}\mu_0}}{(G_0+L_0)^2}. \]

Since $0< G_0< L_0$, then $\kappa ^2 L_0 (G_0+L_0)^3+2(-1)^{j-1}\mu _0>0$ and $(-1)^{j-1}\mu _0 (L_0-G_0) >0$ for $j=1$ and $\mu _0>0$ or $j=2$ and $\mu _0<0$. So, the prograde periodic solutions have multipliers characteristic

\[ 1,1,1+\epsilon \lambda_1^{(j)}+ \mathcal{O}(\epsilon^2),1+\epsilon \lambda_2^{(j)}+ \mathcal{O}(\epsilon^2), \]

and therefore these $S_j$-symmetric periodic solutions are unstable.

On the other, considering equation (3.32) with $G_0<0$ implies that the matrix $A$ takes the form

(3.34)\begin{align} A_R^{(j)}& = \mathbb{J}\left( \frac{\partial^2 \bar{H}}{\partial X_k \partial X_l}\right)_{X=0}\nonumber\\ & \quad =\left( \begin{array}{cc} 0 & \displaystyle L_0^2\pi\left(L_0\kappa^2+\dfrac{2({-}1)^{j-1}\mu_0}{(L_0-G_0)^3}\right) \\ \dfrac{2 \pi L_0 ({-}1)^{j-1}\mu_0 (L_0+G_0)}{L_0-G_0} & 0 \end{array}\right), \end{align}

for $j=1,2$. The eigenvalues of the matrix $A_R^{(j)}$ are given by

\[ \rho_{1,2}^{(j)}={\pm} \frac{ \sqrt{2} \pi L_0^{3/2} \sqrt{({-}1)^{j-1}\mu_0(L_0+G_0)} \sqrt{\kappa ^2 L_0 (L_0-G_0)^3+2({-}1)^{j-1}\mu_0}}{(G_0-L_0)^2}. \]

Since $-L_0< G_0<0$, then $\kappa ^2 L_0 (L_0-G_0)^3+2(-1)^{j-1}\mu _0>0$ and $(-1)^{j-1} \mu _0(L_0+G_0) >0$ for $j=1$ and $\mu _0>0$ or $j=2$ and $\mu _0<0$. So, the eigenvalues $\rho _{1,2}^{(j)}$ are reals and the retrograde periodic solutions have characteristic multipliers

\[ 1,1,1+ \epsilon \rho_{1,2}^{(j)}+\mathcal{O}(\epsilon^2),1- \epsilon \rho_{1,2}^{(j)}+\mathcal{O}(\epsilon^2). \]

Thus, we conclude that the retrograde symmetric periodic solutions are unstable.

3.3 The planar generalized Størmer problem

This problem consists in the study of the dynamics of a charged particle around rotating magnetic planets. More specifically, the generalized Størmer problem describes the dynamics of a dust particle of mass $m$ and charge $q$ orbiting a rotating magnetic planet of mass $M$. The magnetic field of the planet is supposed to be a perfect magnetic dipole of strength aligned along the north–south poles of the planet (see [Reference Iñarrea, Lanchares, Palacián, Pascual, Salas and Yanguas17–Reference Iñarrea, Lanchares, Palacián, Pascual, Salas and Yanguas19] for more details on the formulation of this problem). Moreover, the planet magnetosphere is taken as a rigid conducting plasma which rotates with the same angular velocity $\Omega$ as the planet, in such a way that the charge $q$ is subject to a co-rotational electric field. Furthermore, the gravitational interaction in this model takes into account the non-sphericity of the planet which is given by means of the so-called $J_2$ term. See [Reference Iñarrea, Lanchares, Palacián, Pascual, Salas and Yanguas19] for a complete discussion of the Størmer problem with $J_2$ effect. We will consider the planar generalized Størmer problem which is given through the following two-degree-of-freedom Hamiltonian

(3.35)\begin{align} H(x,y,p_x,p_y)& =\frac{1}{2}(p_{x}^2+p_{y}^2)-\frac{1}{\sqrt{x^2+y^2}}-\frac{\delta (xp_y-yp_x)}{(x^2+y^2)^{3/2}}+ \frac{\delta\beta}{\sqrt{x^2+y^2}}\nonumber\\ & \quad-\frac{J_2}{2(x^2+y^2)^{3/2}} +\frac{\delta^2}{2(x^2+y^2)^2}. \end{align}

It is easy to check that the Hamiltonian function (3.35) is invariant under the two symmetries $S_1$ and $S_2$. The planar generalized Størmer problem (3.35) depends also on three external parameters, namely, $\delta, \beta$ and $J_2$. The parameter $\delta$ indicates the ratio between the magnetic and the Keplerian interaction (i.e., the charge–mass ratio $q/m$ of the particle). The parameter $\beta$ is the ratio between the electrostatic and the Keplerian interactions (i.e., the ratio $\Omega /w_k$, where $w_k=\sqrt {M/\mathcal {R}}$ and $\mathcal {R}$ is the equatorial radius of the planet). Finally, $J_2$ is the oblateness of the planet taken into consideration.

For our purpose, we introduce the small parameter $\epsilon$ by means of the following relations $\delta = \epsilon b$ and $J_2=\alpha \epsilon$ with $b, \alpha$ real numbers. So, the Hamiltonian function (3.35) becomes

(3.36)\begin{align} H& =\frac{1}{2}(p_{x}^2+p_{y}^2)-\frac{1}{\sqrt{x^2+y^2}}\nonumber\\& \quad +\epsilon\left[\frac{-b (xp_y-yp_x)}{(x^2+y^2)^{3/2}}+ \frac{b\beta}{\sqrt{x^2+y^2}}-\frac{\alpha}{2(x^2+y^2)^{3/2}}\right]+\mathcal{O}(\epsilon^2).\end{align}

The next step is to express (3.36) in the mixed coordinates involving polar and Delaunay elements. Then, we arrive at

(3.37)\begin{equation} \mathcal{H}={-}\frac{1}{2L^2}+\mathcal{H}_1(r,\varphi,g,L,G)+\mathcal{O}(\epsilon ^2), \end{equation}

where $(r,\theta )$ are the classical polar coordinates, $\theta =f+g$ and $f$ is the true anomaly. The perturbed function $\mathcal {H}_1$ is given by

(3.38)\begin{equation} {\mathcal{H}}_1={-}\frac{b G}{r^3}+\frac{b\beta}{r}-\frac{\alpha}{2 r^3}={-}\frac{-2 b \beta L^2 (1- e \cos E)^2+\alpha +2 b G}{2 L^2 (1- e\cos E)^3}, \end{equation}

where $r=a(1-e^2)/(1+e\cos f)$ and $e=\sqrt {1-(G^2/L^2)}$ is the eccentricity of the unperturbed elliptic orbit.

To obtain symmetric periodic solutions, we must verify the conditions (a) and (b) in theorem 2.3. We have the following result for the existence of the symmetric periodic solutions of second kind for the generalized Størmer problem (3.36).

Theorem 3.4 Fix the energy level ${\mathcal {H}}_{0}= - \frac {1}{2 L_0^2}$ and the period $T=2\pi L_0^3$ of the elliptic Kepler solution. Given $\alpha$ and $b$ non null real constants, for the $2$-DOF generalized Størmer problem (3.36) for all $\epsilon$ and $\delta L$ positive and sufficiently small, the following statement holds:

If $\alpha b<0$ (resp. $\alpha b> 0$) and $L_0> \frac {3}{4}\left |-\frac {\alpha }{b}\right |$, then there exist two $1$-parameter ($\epsilon$) families and two $2$-parameter ($\epsilon$ and $\delta L$) families of initial conditions such that each of them gives rise to a prograde (resp. retrograde) second-kind $S_i$-symmetric periodic solution.

The $S_1$-symmetric periodic solutions are obtained as continuation of the elliptic Keplerian solution with initial conditions

\[ Y_{0}^{1,k}=\left(0,(2k+1)\frac{\pi}{2},L_0, G_0\right), \quad k=0,1, \]

where $G_0=-\frac {3\alpha }{4b}$. On the other, the $S_2$-symmetric periodic solutions are obtained as continuation of the elliptic Keplerian solution with initial condition

\[ Y_{0}^{2,k}=\left(0,k\pi,L_0, G_0\right),\quad k=0,1. \]

Moreover, the families of symmetric periodic solutions generated by the $1$-parameter families of initial conditions have fixed period $T=2\pi L_{0}^{3}$ and those generated by the $2$-parameter families of initial conditions have period $\overline {T}=T(1- \epsilon T^*)+\mathcal {O}(\epsilon ^2)$ where $T^*=\frac {4}{9} \pi b L_0 (9 \beta -\frac {8 b^2 L_0}{\alpha ^2})$. All these symmetric periodic solutions are close to elliptic Keplerian solutions with eccentricity $e_0=\sqrt {16b^2L_0^2-9\alpha ^2}/4bL_0$.

Proof. Maintaining the notation of § 2, we will verify the hypotheses (a) and (b) of theorem 2.3. For the condition (a), we need to find the solutions of

(3.39)\begin{equation} g^{(1)}(T/2,{\bf Y}^{j,k})\Big |_{{\bf Y}^{j,k}={\bf Y}_{0}^{j,k}}=\int_{0}^{T/2}\frac{\partial {\mathcal{H}}_{1}}{\partial G}(\varphi_{0}(\tau, {\bf Y}^{j,k}))\Big|_{{\bf Y}^{j,k}={\bf Y}_{0}^{j,k}}d\tau=0, \end{equation}

for $j=1$ and $j=2$. Observe that the perturbed function $\mathcal {H}_1$ in (3.38) does not depend on the variable $g$, and therefore, we have $\frac {\partial {\mathcal {H}}_{1}}{\partial G}(\varphi _{0}(\tau, {\bf Y}_{0}^{1,k}))= \frac {\partial {\mathcal {H}}_{1}}{\partial G}(\varphi _{0}(\tau, {\bf Y}_{0}^{2,k}))$ for $k=1,2$. Using expression (2.24) for calculations involving the derivative $\frac {\partial {\mathcal {H}}_{1}}{\partial G}$, we arrive at

(3.40)\begin{align} \displaystyle \frac{\partial {\mathcal{H}}_{1}}{\partial G}& = \displaystyle \frac{1}{4 a^4 e (e \cos E-1)^5} \left[\cos E \left(a^2 b \beta \left(11 e^2+4\right) G-8 a b e^2-6 G (\alpha +2 b G)\right)\right.\nonumber\\ & \quad\left. +\,e\left(4ab+2 a b e^2 -2 a^2 b \beta e^2 G-8 a^2 b \beta G-2 a b \cos (2 E) \left(e^2 (a \beta G-1)\right.\right.\right.\nonumber\\ & \quad\left.\left.\left. +\,2 a \beta G\right)+12 b G^2+6 \alpha G+a^2 b \beta e G \cos (3 E)\right)\right]. \end{align}

where $a=L^2$. Next, we evaluate (3.40) on the solution $\varphi _{0}(t, {\bf Y}_{0}^{j,k})$. It follows that

(3.41)\begin{align} g^{(1)}(T/2,{\bf Y}_0^{j,k})& =\displaystyle\int_{0}^{T/2}\frac{\partial {\mathcal{H}}_{1}}{\partial G}(\varphi_{0}(\tau, {\bf Y}_0^{j,k}))\,{\rm d}\tau\nonumber\\ & = \int_{0}^{\pi}\frac{\partial {\mathcal{H}}_{1}}{\partial G}(\varphi_{0}(\tau, {\bf Y}_0^{j,k}))L_0^3(1-e_0\cos E)\,{\rm d}E, \end{align}

Therefore, after integration of (3.41) we arrive at

(3.42)\begin{equation} g^{(1)}(T/2,{\bf Y}_0^{j,k})=\frac{ L_0^3 \left(2 a b \left( e_0^2-1\right)+3 G_0 (\alpha +2 b G_0)\right)}{2 a^4 \left(1-e_0^2\right)^{5/2}}=\frac{\pi (3 \alpha +4 b G_0)}{2 G_{0}^3 | G_0| }. \end{equation}

We must find the solutions $G_0$ of the $g^{(1)}(T/2,{\bf Y}_0^{j,k})=0$, in the two domains given in (3.23). Observe that in both domains, the solution of the equation $g^{(1)}(T/2,{\bf Y}_0^{j,k})=0$ is given by

(3.43)\begin{equation} G_0={-}\frac{3\alpha}{4b}. \end{equation}

Thus, in the region $D_1$ (prograde solutions), we need to impose that $\alpha b<0$ and $L_0>-\frac {3 \alpha }{4 b}$. While, in the region $D_2$ (retrograde solutions), we must have $\alpha b>0$ and $L_0>\frac {3 \alpha }{4 b}$.

Now, we verify the condition (b) of theorem 2.3. To compute the partial derivative $\frac {\partial }{ \partial \delta G}\left (\frac {\partial \mathcal {H}_1}{ \partial G}\right )$, we use the chain rule and expressions (2.24) to differentiate (3.40) with respect to $\delta G$. Thus, again using the change of variables $t=L_{0}^{3}(E-e_0\sin E)$, after integration, we obtain

(3.44)\begin{equation} \frac{\partial g^{(1)}}{\partial \delta G}\Big |_{{\bf Y}^{j,k}\!=\!{\bf Y}_{0}^{j,k}}=\int_{0}^{\pi}\frac{\partial }{ \partial \delta G}\left(\frac{\partial \mathcal{H}_1}{ \partial G}\right)\Big |_{{\bf Y}^{j,k}={\bf Y}_{0}^{j,k}}L_0^3(1\!-\!e_0\cos E)\,{\rm d}E\!={-}\frac{6 \pi (\alpha +b G_0)}{G_0^4 |G_0|}. \end{equation}

Evaluating expression (3.44) on the solutions $G_0=-\frac {3 \alpha }{4 b}$, we get

\[ \frac{\partial g^{(1)}}{\partial \delta G}\Big |_{{\bf Y}^{j,k}={\bf Y}_{0}^{j,k}}=\frac{512 \pi b^5}{81 \alpha ^4}\neq 0, \]

since $b\neq 0$. Therefore, we have verified the conditions given in (2.12). Thus, we conclude that every Keplerian elliptic orbit with initial condition

\[ {\bf Y}_{0}^{j,k}= \left(0, g_{0}^{j,k}, L_0,\pm G_0\right),\quad G_0={-}\frac{3\alpha}{4b}, \quad k=1,2, \]

can be continued to a $S_j$-symmetric periodic solution of the generalized Størmer problem. Therefore, we conclude the proof.

We point out that in the previous theorem, we cannot give information about the linear stability of the symmetric periodic solutions, because the matrix $A$ (2.11) is given by $\left (\begin {array}{cc} 0 & * \\ 0 & 0 \\ \end {array}\right )$, so their eigenvalues are all null.

Remark 3.5 A simple calculation shows that the averaged function of (3.38) in Delaunay variables is

\[ \overline{\mathcal{H}}=\frac{-\alpha +2 b \beta L | G| ^3-2 b G}{2 L^3 | G| ^3}. \]

Moreover, the averaged system associated with the Hamiltonian (3.36), for prograde solutions is given by

(3.45)\begin{equation} \begin{aligned} \frac{{\rm d} G}{{\rm d} \ell} & ={-}\epsilon \frac{1}{({-}2h^*)^{3/2}}\frac{3 \alpha +4 b \text{G}}{2 \text{G}^4 \text{L}^3},\\ \frac{{\rm d} g}{{\rm d} \ell} & =0. \end{aligned} \end{equation}

Therefore, system (3.45) has no non-degenerate critical points. The same conclusion is valid for retrograde solutions, and Reeb's Theorem or Averaging theory does not give us information about the existence of periodic solutions.