Given the following equation:
$$ \dot{\Theta} = \operatorname{sgn}(z) \sqrt{| z |} $$
$$ z = \dot{\Theta}^2 \operatorname{sgn}(\dot{\Theta}) $$
$$ \dot{z} = 2 \sqrt{| z |} \ddot{\Theta} $$
how can I get the vector field $\boldsymbol{g}$?
$$ \dot{\boldsymbol{q}} = \boldsymbol{g} (\boldsymbol{q}) $$
where $\boldsymbol{q} = (\Theta, z)^T$.
This expression was taken from the scientific paper "Dynamics and stability of a rimless spoked wheel: a simple 2D system with impacts" by Michael J. Coleman.
EDIT: another equation which is in the paper is:
$$ \ddot{\Theta} - \sin(\Theta + \alpha) = 0 $$
Equation $(7)$ specifies the time evolution of each of the generalized coordinates by providing $\dot{\theta}(\theta,z)$ (which actually has no explicit $\theta$ dependence in this case) and $\dot{z}(\theta,z)$. The idea of equation $(9)$ is to repackage these two scalar equations as one vector equation for the time evolution of the state vector $\boldsymbol{q}=\begin{bmatrix} \theta & z\end{bmatrix}^{\intercal}$. This is done by defining the vector field $\boldsymbol{g}: \boldsymbol{q}=\begin{bmatrix} \theta & z\end{bmatrix}^{\intercal} \mapsto \boldsymbol{\dot{q}} = \begin{bmatrix} \dot{\theta}(\theta,z) & \dot{z}(\theta,z) \end{bmatrix}^{\intercal}$. You can see the role of $\boldsymbol{g}$ is to map the state vector $\boldsymbol{q}$ to its temporal rate of change $\boldsymbol{\dot{q}}$ such that $\boldsymbol{\dot{q}}=\boldsymbol{g}(\boldsymbol{q})\:(9)$ and that this is done component-wise by viewing $\boldsymbol{\dot{q}} = \begin{bmatrix} \dot{\theta}(\boldsymbol{q}) & \dot{z}(\boldsymbol{q}) \end{bmatrix}^{\intercal}$.