Why can the coriolis force potential be written as $ E_\text{cor}=m\dot{\theta}\begin{vmatrix} X & Y \\ \dot{X} & \dot{Y} \end{vmatrix}$? I found the following formula for the Coriolis force written here:
$$ E_\text{cor}=m\dot{\theta}\begin{vmatrix}
X & Y \\ 
\dot{X} & \dot{Y}
\end{vmatrix}=m\dot{\theta}\ (\dot{Y}X-\dot{X}Y)$$
I have the following questions:

*

*Does this matrix have a certain name? Is it related to Jacobian or Hessian?
$\begin{vmatrix}
X & Y \\ 
\dot{X} & \dot{Y}
\end{vmatrix}$


*Where does it come from?


*Also this seems like a commutator relation: $(\dot{Y}X-\dot{X}Y)$, is that true?


*Can a similar matrix be used for the centrifugal force?
 A: This can be seen as a consequence of the fact that we can write the triple product $\vec{v} \cdot (\vec{\Omega} \times \vec{r})$ as a determinant, rearrange it, and then do an expansion by subminors:
\begin{align*}
\vec{v} \cdot (\vec{\Omega} \times \vec{r}) &= \begin{vmatrix} v_x & v_y & v_z \\ \Omega_x & \Omega_y & \Omega_z \\ x & y & z \end{vmatrix} \\
&= \begin{vmatrix} \Omega_x & \Omega_y & \Omega_z \\ x & y & z  \\ v_x & v_y & v_z \end{vmatrix} \\
&= \Omega_x \begin{vmatrix} y & z \\ v_y & v_z \end{vmatrix} - \Omega_y \begin{vmatrix} x & z \\ v_x & v_z \end{vmatrix} + \Omega_z \begin{vmatrix} x & y \\ v_x & v_y \end{vmatrix}
\end{align*}
In the case where $\Omega_x = \Omega_y = 0$ and $\Omega_z = \dot{\theta}$, this reduces to the formula you found.
As far as whether this can be used for the centrifugal potential, that's of the form $(\vec{\Omega} \times \vec{r}) \cdot (\vec{\Omega} \times \vec{r})$, and so the trick above does not work so elegantly (since it's not a vector triple product like the Coriolis term is.)
A: the potential energy for Coriolis force is:
$$U=-m\,\vec v\cdot (\vec \Omega\times \vec r)$$
with
$$\vec r=\begin{bmatrix}
  {x} \\
  {y} \\
  {z} \\
\end{bmatrix}\quad,
\vec v=\begin{bmatrix}
  \dot{x} \\
  \dot{y} \\
  \dot{z} \\
\end{bmatrix}\quad,
\vec \Omega=\dot\theta
\begin{bmatrix}
  0 \\
  0 \\
  1 \\
\end{bmatrix}\quad\Rightarrow$$
$$U=m\,\dot\theta\,(y\,\dot x-x\,\dot y)=m\,\dot\theta\,\det(\mathbf A)\\
\mathbf A=\begin{bmatrix}
   y & x \\
   \dot{y} & \dot{x} \\
 \end{bmatrix}
$$
nothing to do with determinate , this is spatial  case because
$~\vec\Omega=\dot\theta\vec e_z$ ?
