# Some Dynamic Aspects of a Sextic Galactic Potential in a Rotating Reference Frame
## 1 Introduction  
Galactic dynamics, the study of the movements of galactic masses, is an important branch of astrophysics. It includes the investigations of stars, gases, and dark matter with the goal of understanding the main kinematics features of the three main types of galaxies (elliptical, spiral, and irregular) [1; 2; 3; 4]. The theoretical bases for the subject were established approximately seven decades ago, when the study of dynamics was dominated by integrable and near-integrable systems. The study of galactic motion can be carried out globally. It can also be performed locally near the equilibrium points by using the perturbed oscillators. In the last few decades, several studies on galactic dynamics, mostly focused on elliptical galaxies, were published [5; 6; 7; 8; 9; 10; 11; 12], and the existence of periodic solutions and their stability were investigated [13; 14; 15; 16; 17; 18; 19]. [PERSON] et al. [20] also investigated the bifurcations of the Armbruster-Guckenheimer-Kim (AGK) potential by studying the topological type of the level sets of the integrable cases of the AGK galactic potential. This study was followed by [PERSON] et al. [21], who analyzed its global dynamics. Except for some studies on stellar orbits, the rotation of galaxies has been ignored in most studies of galactic dynamics [22]. This is in spite of the fact that an elliptical galaxy rotates with a slight angular velocity, and the force produced by the galaxy's rotation will influence the dynamic (see, for example, [23; 24]). This motivates the study of the rotation of a galaxy, modeled by a bisymmetric potential with different perturbations, and the effects of this rotation on the dynamics of star motion. [PERSON] et al. studied the stability of the motion equilibrium positions for a Henon-Heiles system in a rotating reference frame [25]. [PERSON] in [26] studied the dynamic features of the AGK potential in a rotating reference frame. In that study, he examined the nonintegrability of the system toseek periodic solutions near the equilibrium points. [PERSON] et al. [27] investigated the integrability and stability of equilibria for a quartic galactic potential in a rotating reference frame. [PERSON] et al. [28] applied [PERSON]'s theorem to prove the existence of periodic orbits in a rotating Henon-Heiles system. In the present work, we consider the plane motion rotation with a constant angular velocity \(\Omega\) of a galaxy approximately symmetrical around an axis normal to the plane of rotation. By reversing the normal axis if needed, we may assume that \(\Omega\geq 0\). This motion, similar to that of most elliptical galaxies, is described by a Hamiltonian system that involves a symmetric potential. The Hamiltonian we use is given by  
\[H(x,y)=\frac{1}{2}(p_{x}^{2}+p_{y}^{2})-\Omega(xp_{y}-yp_{x})+U(x,y), \tag{1}\]  
where \(p_{x},p_{y}\) are the generalized momentum corresponding to the generalized coordinates \((x,y)\) and \(U\) is the four-parameter galactic potential function given by  
\[U(x,y)=\frac{1}{2}(x^{2}+y^{2})+\frac{1}{6}(ax^{6}+by^{6})+\frac{1}{2}x^{2}y^{ 2}(cx^{2}+dy^{2}), \tag{2}\]  
while \(a,b,c\), and \(d\) are free constants. Recently, [PERSON] et al. [29] studied the existence of periodic solutions and their stability by applying the averaging theorem for the sextic galactic potential while ignoring the rotation of the galaxy. This motivated us to study the influence of the galaxy rotations on some dynamics of the motion which were not considered before in the literature, such as integrability, the stability of the equilibrium positions, and the existence of periodic solutions near the equilibrium positions. The Hamilton canonical equations are  
\[\dot{x} =p_{x}+\Omega y, \tag{3a}\] \[\dot{y} =p_{y}-\Omega x,\] (3b) \[\dot{p}_{x} =\Omega p_{y}-x\Big{[}1+ax^{4}+2 cx^{2}y^{2}+dy^{4}\Big{]},\] (3c) \[\dot{p}_{y} =-\Omega p_{x}-y\Big{[}1+by^{4}+2 dx^{2}y^{2}+cx^{4}\Big{]}. \tag{3d}\]  
As usual, the dots indicate that the derivative expressed is with respect to time.