I'm trying to prove the following:
Let $X=(-y)\frac{\partial}{\partial x}+(x)\frac{\partial}{\partial y}$ and $\omega=Vdx\wedge dy$ be a smooth volume element in $\mathbb{R}^2$. Show that $L_X\omega=0$ if and only if $V(x,y)$ depends only on $x^2+y^2$.
Intuitively I know why is this true: Integral curves of $X$ are actually circles around origin and $L_X\omega=0$ means that $\omega$ is constant on integral curves so $V$ is a constant multiple of $x^2+y^2$. Actually I can prove that $\Phi_t(\omega)=\omega$ for all possible $t$ where $\Phi$ is the flow of $X$.
I have problem writing down the proof since I do not know how to find the flow of $X$ and finish the proof.
Any help is appreciated.