I have come across an interesting property of a dynamical system, being transformed by a map, but i haven't been able to figure out why this is happening (for quite some time now actually). Any help is greatly appreciated. Here goes then:

Let M be a n-D manifold and $\dot x=F(x)u_1, F\in \mathbb{R}^{n\times m}, x \in \mathbb{R}^{n}, u_1 \in \mathbb{R}^{m}$ be a control system evolving on M (F is the system matrix i.e. state transition function, and $u_1$ is the input of the system. For all practical purposes $u_1$ is an m-vector from an input space $\mathbb{R}^{m}$). Now let $x=\Psi (y)$ be a coordinate change on M and $u_2=M(y)u_1$ a transformation of the input $u_1$ of the first system. By applying these maps on the system, you get the new equations  $\dot y=F(y)u_2$. As you may notice, F is the same in both systems. The problem is why is this happening i.e. for what systems and transformations does this property hold?

A little more elaboration

It is useful to investigate the maps more closely. In the general case one has

$\dot x=D\Psi \dot y$
$\dot x= F(x)u_1$

thus
$\dot y=D\Psi ^{-1} F(x)u_1$, (1)

where $D\Psi$ is the Jacobian matrix of $\Psi$. In our case it actually turns out that:  

$\dot y=F(y)M(y)u_1$. (2)

You can then consider that $u_2=M(y)u_1$ and get the final system,

$\dot y=F(y)u_2$,

that is, the same system.
 By (1),(2) you get,

$D\Psi ^{-1} F(x)u_1=F(y)M(y)u_1 \Rightarrow (D\Psi ^{-1} F(x)-F(y)M(y))u_1=0$. 

Since this holds for every $u_1$, you have the condition,

$F(\Psi (y))=D\Psi F(y)M(y)$

So, what does this condition imply? What systems F and maps $\Psi$ hold this property (of system invariance)? I should note that F is nonlinear and a case study where this actually happens is the kinematic model of a unicycle robot i.e. this. Any ideas?


Question: God the Father's possession of a body of flesh and bones
Answer: The OT is very clear that any static image or representation of the Divine is a very bad thing. The first two commandments are prohibitions of idolatry and the primary charge of the Prophets is against the same. To ascribe flesh and image to God would be heretical to the Jewish writers and authors of the books. As such, using a "primary sense to the primary recipient" hermeneutic would completely fail. Idolatry was so antithetical to the Jewish understanding would have made giving God any flesh or image heretical to unthinkable. 

That said (and I say this as a non-Mormon), the primary problem and setting in which Joseph Smith either wrote or was called in (depending on your belief), was not one in which idolatry was an issue. As such, it is difficult to apply a Jewish understanding to a Mormon prophet. Would a Jew think this silly? Sure. But then again, the idea that God would ever take on flesh even temporarily as the Messiah is equally repugnant, so as an argument there is not much to gain.