# Some Dynamic Aspects of a Sextic Galactic Potential in a Rotating Reference Frame
## 5 Periodic Motion  
In this section, periodic solutions for the Hamiltonian system in Equation (3) around its equilibrium positions are discussed based on the Lyapunov theorem [42], which has been successfully applied in several works. [PERSON] [43] presented the periodic solutions near the equilibrium position for mechanical systems in general, while [PERSON] introduced periodic solutions for the motion of a rigid body's dynamics in a central Newtonian field [44]. [PERSON] [26] found periodic solutions for the AGK potential nearest to its equilibrium points. [PERSON] et al. [29] constructed periodic solutions for the sextic galactic potential in a non-rotating reference frame. The Lagrangian equations of the motions are  
\[\ddot{x}-2\Omega\dot{y}+V_{x}=0,\qquad\ddot{y}+2\Omega\dot{x}+V_{y}=0, \tag{34}\]  
where \(V\) is given by Equation (19). The equations of the motion in Equation (34) has a first integral  
\[\frac{1}{2}(\dot{x}^{2}+\dot{y}^{2})+V=E, \tag{35}\]  
where \(E\) is an arbitrary constant. This integral is called the Jacobi integral. Since the equilibrium point \((x_{0},y_{0})\) is a critical point for the effective potential \(V\), we have  
\[V_{x}(x_{0},y_{0})=V_{y}(x_{0},y_{0})=0,\qquad E_{0}=V(x_{0},y_{0}), \tag{36}\]  
where \(E_{0}\) denotes the Jacobi integral value at the point \((x_{0},y_{0})\). To describe the perturbed motion about \((x_{0},y_{0})\), we introduce  
\[x=x_{0}+p,\qquad y=y_{0}+q, \tag{37}\]  
where \(p\) and \(q\) are perturbation variables. Notice that the expressions in Equation (37) transform the equilibrium point \((x_{0},y_{0})\) to the origin of the coordinates. When inserting the expressions in Equation (37) into the equations of motion (34), we have  
\[\ddot{p}-2\Omega q+V_{xx}(x_{0},y_{0})p+V_{xy}(x_{0},y_{0})q=0,\qquad\ddot{q}+ 2\Omega\dot{p}+V_{xy}(x_{0},y_{0})p+V_{yy}(x_{0},y_{0})q=0. \tag{38}\]  
Figure 3: Regions of stability and instability for the equilibrium position \(\mathcal{E}_{5,6}\) on the plane of the parameters \(c\) and \(a\). Using a new variable \(\zeta\) instead of the time through the time transformation yields  
\[\zeta=\gamma t, \tag{39}\]  
where \(\gamma\) is a free parameter. By applying the transformation in Equation (39), the system in Equation (38) is converted into  
\[ \begin{split}&\gamma^{2}p^{\prime\prime}-2\gamma\Omega q^{\prime}+V_{ xx}(x_{0},y_{0})p+V_{xy}(x_{0},y_{0})q=0,\\ &\gamma^{2}q^{\prime\prime}+2\gamma\Omega p^{\prime}+V_{xy}(x_{0},y_{0})p+V_{yy}(x_{0},y_{0})q=0,\end{split}  \tag{40}\]  
where \({}^{\prime}\) refers to the derivative with respect to \(\zeta\). Following the [PERSON] theorem [42], we assume the periodic solutions as a power series of \(n\) which, in our case, depends on the Jacobi integral value (i.e., it depends on \(E\)). Assume that  
\[p=\sum_{j=1}^{\infty}n^{j}x_{j},\quad q=\sum_{j=1}^{\infty}n^{j}y_{j},\quad E= E_{0}+\sum_{j=2}^{\infty}n^{j}E_{j}, \tag{41}\]  
where \(E_{i}\) are constants and \(x_{i}\) and \(y_{i}\) are \(T-\) periodic functions of \(t\) with a period \(\gamma\) that can be written as  
\[T=\frac{2\pi}{\gamma}=\frac{2\pi}{\gamma_{0}}\Bigg{(}1+\sum_{j=2}^{\infty}n^{ j}T_{j}\Bigg{)}. \tag{42}\]  
By substituting the expressions in Equations (41) and (42) into Equation (40) and finding the first order approximation, we have  
\[ \begin{split}&\gamma_{0}^{2}x_{1}^{\prime\prime}+2\gamma_{0} \Omega y_{1}^{\prime}+V_{xy}(x_{0},y_{0})y_{1}+V_{yy}(x_{0},y_{0})x_{1}=0,\\ &\gamma_{0}^{2}y_{1}^{\prime\prime}-2\Omega\gamma_{0}x_{1}^{ \prime}+V_{xx}(x_{0},y_{0})y_{1}+V_{xy}(x_{0},y_{0})x_{1}=0.\end{split}  \tag{43}\]  
Our goal is accomplished by looking for solutions to Equation (43) in the form of  
\[ \begin{split}& x_{1}=a_{0}+a_{1}\cos\zeta+a_{2}\sin\zeta,\\ & y_{1}=b_{0}+b_{1}\cos\zeta+b_{2}\sin\zeta,\end{split}  \tag{44}\]  
where \(a_{i},b_{i},i=0,1,2\) are the free constants that need to be determined. When putting the expressions in Equation (44) into Equation (43), we obtain  
\[