# Some Dynamic Aspects of a Sextic Galactic Potential in a Rotating Reference Frame
## 2 Integrability Analysis  
A 2D Hamiltonian system is integrable if it possesses two functionally independent integrals of motion \(J_{1}\) and \(J_{2}\) that are in involution (i.e., \(\{J_{1},J_{2}\}=0\)), where \(\{.,.\}\) representsPoisson brackets [30]. Integrable systems exhibit regularity, enabling the prediction of their behavior for all times. Their explicit solution can be found using quadratures, and they can be used to study non-integrable systems through perturbation theories [31]. The Hamiltonian system in Equation (1) is integrable if it has a complementary integral independent of the Jacobi integral in Equation (4). The problem of finding this integral is complicated due to the fact that its functional form is unknown. Details on constructing integrals of motion are mentioned in [32; 33; 34; 35; 36]. On the other hand, there are many methods that prove non-integrability, such as the Yoshida and Morales-Ramis criteria, which are based on the investigation of normal variation equations along a particular solution, and the Ziglin approach, which depends on the examination of the monodromy group associated with a variational equation [37; 38; 39]. Neither of these techniques work with Hamiltonian systems with linear momentum terms. As a consequence, we rewrote the Hamiltonian in Equation (1) appropriately. We use the transformation  
\[t\to\alpha^{-2}t,\quad(x,y)\to(\alpha x,\alpha y),\quad(p_{x},p_{y})\to(\alpha ^{3}p_{x},\alpha^{3}p_{y}). \tag{5}\]  
Then, the Hamiltonian equations (Equation (3)) are transformed into  
\[\dot{x} =p_{x}+\frac{\Omega}{\alpha^{2}}y, \tag{6a}\] \[\dot{y} =p_{y}-\frac{\Omega}{\alpha^{2}}x,\] (6b) \[\dot{p}_{x} =\frac{\Omega}{\alpha^{2}}p_{y}-\frac{1}{\alpha^{4}}x-x[ax^{4}+2 cx^{2}y^{2}+dy^{4}],\] (6c) \[\dot{p}_{y} =-\frac{\Omega}{\alpha^{2}}p_{x}-\frac{1}{\alpha^{4}}y-y[by^{4}+ cx^{4}+2 dx^{2}y^{2}]. \tag{6d}\]  
When \(\alpha\) tends toward infinity, the non-similarity system in Equation (6) becomes  
\[\dot{x} =p_{x},\qquad\dot{y}=p_{y}, \tag{7a}\] \[\dot{p}_{x} =-x[ax^{4}+2 cx^{2}y^{2}+dy^{4}],\] (7b) \[\dot{p}_{y} =-y[by^{4}+cx^{4}+2 dx^{2}y^{2}]. \tag{7c}\]  
The system in Equation (7) is a canonical Hamilton equation derived from the Hamiltonian  
\[H_{0}=\frac{1}{2}(p_{x}^{2}+p_{y}^{2})+\frac{1}{6}(ax^{6}+by^{6})+\frac{1}{2} x^{2}y^{2}(cx^{2}+dy^{2}). \tag{8}\]  
We will study the non-integrability of the system in Equation (8) and show that the non-integrability of the Hamiltonian \(H_{0}\) implies the non-integrability of the Hamiltonian \(H\) in Equation (1):  
Theorem 1: _If the Hamiltonian \(H_{0}\) is non-integrable, then the Hamiltonian \(H\) is non-integrable._  
Proof: If the Hamiltonian \(H\) is integrable, then it has a complementary integral \(J_{2}=J_{2}(x,y,p_{x},p_{y})\) independent of the Jacobi integral in Equation (4). By applying the transformation in Equation (5) to the complementary integral \(J_{2}\) and taking the limit as \(\alpha\) tending toward infinity, we obtain a complementary integral for \(H_{0}\), confirming the integrability of \(H_{0}\). The non-integrability of \(H\) follows by contradiction. [PERSON] introduced a non-integrability theorem based on [PERSON]'s analysis [37]. This theorem deals only with homogeneous potentials:  
**Theorem 2**. : _([PERSON]'s theorem) Let \(H_{0}=\frac{1}{2}(p_{x}^{2}+p_{y}^{2})+V_{0}(x,y)\) be a Hamiltonian system with homogeneous potential \(V_{0}\) of a degree \(k\geq 3\). Then, it is non-integrable if_  
\[\sigma\in]-\infty,0[\ \underset{j\in\mathbb{N}}{\cup}\ ]j(j-1)\frac{k}{2}+j,j(j+1) \frac{k}{2}-j[, \tag{9}\]  
_where_  
\[\dot{x} =p_{x}+\frac{\Omega}{\alpha^{2}}y, \tag{6a}\]0  
_is the non-integrability coefficient and \(\mathbf{c}\) is the solution of the algebraic equation \(\
abla V_{0}(\mathbf{c})=\mathbf{c}\)._  
We are going to apply Theorem 2 to investigate the non-integrability of the Hamiltonian \(H_{0}\), which is given by Equation (8).