I'm learning differential equations and trying to solve the following problem:
$x_1'=x_2^2$
$x_2'=x_4+x_3$
$x_3'=x_4^2$
$x_4'=x_1-x_2+x_3^2$
We want to determine whether the system of equations is Hamiltonian.
I verified the necessary condition, which is:
$\frac{\partial fj}{\partial xj}=-\frac{\partial fi}{\partial xi}$ for each $i \neq j$.
I am looking for pairs but the $\frac{\partial fj}{\partial xj}=0$ for every $j$. I'm not sure how to continue in this case.
Could you please help me?