I'm not entirely familiar with viewing differential equations from a differential geometry viewpoint.
As far as I understand it, in terms of differential geometry, polar coordinates are a 2-dimensional manifold over the set $ M = \mathbb{R}_{>0} \times \mathbb{S}^1$.
One way to define a topology on $M$ would be to take all the disks $$ D_R = \{(r, \theta) \in M : r < R\}\\ \mathcal{D} = \bigcup_{R\in \mathbb{R}_{> 0}} D_R. $$
There exists one global chart $(U, \varphi)$ with $U = M, V = \mathbb{R}^2\setminus\{0\}$ and $\varphi: U \to V $, usually called the "polar coordinate transformation" map a point $p = (r, \theta)$ on the manifold to coordinates $x \in \mathbb{R}^2$ $$ x^1 =r \cos(\theta)\\ x^2 = r \sin(\theta), $$ which necessarily (as a chart) is continuous and invertible and the inverse is defined on $\varphi^{-1}: V \to U$. (I understand, that if I prove additional properties of this chart, like e.g. smoothness, then that shows $\mathcal{M}$ to be a smooth manifold.)
Let us call $\mathbb{R} = (R, \mathcal{O}, \mathcal{A})$ the standard manifold on $\mathbb{R}^2$ with default topology and an atlas consiting only on the identity.
For my current problem, I am given a first oder differential equation defined by a vectorfield $\{\xi(x)\}_{x\in \mathcal{R}}$ in terms of (coordinate functions ?) $$ \xi(x) = \begin{bmatrix}\xi^1(x)\\\xi^2(x)\end{bmatrix}, $$ as $$ \begin{bmatrix} \dot{x}^1\\ \dot{x}^2 \end{bmatrix} = \begin{bmatrix} \xi^1(x)\\ \xi^2(x) \end{bmatrix} $$
Now unfortunately, the component function of $\xi(x)$ are not "nicely shaped" and make it hard to state precicely some properties I am after. So what I did was taking the inverse $\varphi^{-1}$ of the aforementioned chart and taking the time derivative, the subsequently substituting in $\varphi^1(p), \varphi^2(p)$. My differential equation now looks like this: $$ \begin{bmatrix} \dot{r}\\ \dot{\theta} \end{bmatrix} = \begin{bmatrix} f^1(p)\\ f^2(p) \end{bmatrix}. $$
Now, in the terminology of differential geometry, what did I do here?
Moreover, $\mathcal{R}$ can be endowed with a norm as a metric, so that I can show $\xi(x)$ to be Lipschitz, which is direly needed for the existence of solutions. However, I need to be able to check the Lipschitz property directly in polar coordinates.
Am I getting the terminology right so far?
Can somebody guide me how I should proceed to understand this?