I know the definition of a Lie derivative, but I'm having trouble applying that to a concrete example.
The Lie derivative on vector field $X$ and form $\omega$ is $$L_X\omega(p) = \lim_{t\rightarrow 0} \frac{(\phi_t^X)^*(\omega(\phi_t^X(p))) - \omega(p)}{t}$$
So, say I have the vector field $X = -x_2\partial_{x_1} + x_1\partial_{x_2}$, and 1-form $\omega = \frac{x_2dx_1 - x_1dx_2}{x_1^2 + x_2^2}$ (all in $\mathbb{R}^2$), then what is $L_X\omega$, using the limit definition of Lie derivative?
Here, the integral curve of $X$ at a point $p = (p_1, p_2)$ is $(p_1\cos(t) - p_2\sin(t), p_2\cos(t) + p_1\sin(t)$.
I would appreciate it if you could walk me through how to find the Lie derivative, this is not a homework question, but an example that I came up with by picking out arbitrary values that I've came across on this topic. I can't guarantee that this will work, and please suggest an example for me if there is something more illustrative.