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: How much is too much damage?
Answer: "The owner has repaired the damage" does not sound reassuring.  Make an appointment with a luthier and ask to lend the guitar out for testing.

Basically there are several questions to ask:


How likely is the guitar to degrade without additional repairs? Over which time span?
This is important because it determines what you need to invest now in order not to make this a waste of money. 
Which repairs would be warranted for securing its state and sound
quality?
This is important for figuring out how much will get dunked into this instrument before it will hold up for a decade or so.
Which repairs would be warranted for getting good looks back?
This will basically only become relevant when your husband falls in love with the instrument to a degree where he would consider a body job for it, and then this should not become similarly expensive as a new one would. 


You really need to get a verdict from a luthier to figure out whether this instrument will hold up and is not substantially worse in its structure due to the damage.

If it's reasonably salvaged, then a $3000 guitar should be quite more enjoyable to play with than a $300 guitar.

But you are not likely to figure this out on your own (and the price ranges of several followup options), and from the description alone, nobody here is going to either.