I'm a engineering student (i.e. no solid foundations on "true" mathematics), sorry if my question is silly. When I was computing the Jacobian to study the stability of equilibria points on power systems (using the swing equation), I realized that if we exchange the order on which variable we take the derivatives (i.e. the column orderding of the Jacobian) we have different eigenvalues (as we have different matrices) and hence different (in)stabilities. So how do we set the "proper" ordering of variables? I couldn't find anything about this. If anyone is curious, the systems looks like something more or less like this:
$$ f_1 \equiv \frac{\partial\delta}{\partial t} \ = \ \omega \\ f_2 \equiv\frac{\partial^2\delta}{\partial t^2} \ = \ \frac{P_{mec} - P_{elet}sin(\delta) - \xi }{M} $$
The state variables are delta and omega. The $P_i$ , M and $\xi$ are real constants.
The book says that the "proper" ordering is delta and then omega (left and right columns of the Jacobian), but doesn't explain why.
Thanks in advice.