Consider the Hamiltonian $$ H=H(x,y,p_x,p_y) $$ which generates the dynamical system $$ \dot{x}=+\frac{\partial H}{\partial p_x} $$ $$ \dot{y}=+\frac{\partial H}{\partial p_y} $$ $$ \dot{p_x}=-\frac{\partial H}{\partial x} $$ $$ \dot{p_y}=-\frac{\partial H}{\partial y} $$ I then discover that this system admits a certain fixed point $$ \vec{z}_0:=(x_0,\,y_0,\,p_{x,0},\,p_{y,0}) $$ To determine the stability properties of $\vec{z}_0$, I compute the Jacobian matrix $J$ associated to the dynamical system, I evaluate it at fixed point $\vec{z}_0$, and I compute the eigenvalues, which are of the type: $$ \lambda_1=+i\omega_a $$ $$ \lambda_2=-i\omega_a $$ $$ \lambda_3=+i\omega_b $$ $$ \lambda_4=-i\omega_b $$ Therefore, I can recognize the presence of two characteristic frequencies, namely $\omega_a$ and $\omega_b$.
So far, so good.
At this point, I try to do the same computation with a different set of dynamical variables, for example making use of polar coordinates instead of cartesian coordinates. So, I start from Hamiltonian $$ H^\prime=H^\prime(r,\theta,p_r,p_\theta), $$ I find the same fixed point found before, but now written in polar coordinates, i.e. $\vec{z}_0^\prime=(r_0,\theta_0,p_{r,0},p_{\theta,0})$. I then compute the Jacobian matrix $J^\prime$ associated to $H^\prime$, evaluate it at fixed point $\vec{z}_0^\prime$ and compute its eigenvalues. The latter have the following structure: $$ \lambda_1^\prime=+i\omega_a^\prime $$ $$ \lambda_2^\prime=-i\omega_a^\prime $$ $$ \lambda_3^\prime=+i\omega_b^\prime $$ $$ \lambda_4^\prime=-i\omega_b^\prime $$ So, i can recognize two characteristic frequencies: $\omega_a^\prime$ and $\omega_b^\prime$.
What I find really counterintuitive is that $$ \omega_a\neq\omega_a^\prime $$ $$ \omega_b\neq\omega_b^\prime $$ that means that, the two eigenfrequencies are different in the two schemes (the cartesian one and the polar one) that I developed. I would have expected them to be the same, i.e. that $\omega_a=\omega_a^\prime$ and $\omega_b=\omega_b^\prime$.
So, here my question comes: is it possible that different set of dynamical variables lead to different characteristic frequencies for a certain fixed point?
To be noted: actually the two sets of eigenfrequencies, i.e. $\{\omega_a, \omega_b\}$ and $\{\omega_a^\prime, \omega_b^\prime\}$ include similar terms and one can obtain the frequencies in the first set by properly combining the ones in the second set, and viceversa. But, as I said, I expected the two sets to be really identical.