Generalised coordinates I am working on a scientific project for my university and I am reading a german paper (Karas: "Platten unter seitlichem Stoß") which makes use of generalised coordinates. 
It's about an analytical solution of the impact of a ball on a plate.
At some point, he defines the potential energy for the plate $(a,b)$ as:
$$\tag{m,n = 1,2,3,...}V = ... \sum_m \sum_n q_{mn}^2 \left(\frac{m^2}{a^2} + \frac{n^2}{b^2} \right)^2$$
Aswell as the kinetic energy:
$$T = ... \sum_m \sum_n \dot{q}_{mn}^2$$
It doesnt explain what the $q$ stands for. It seems to be some kind of generalised coordinate but when looking for this on the web myself, I didnt find a good explanation.
So my question is:

What does $q_{mn}$ mean? Why can $m,n$ be replaced with any positive number? How could I think about the coordinate as one on the plate? Does it equal the deflection?

 A: The $q_{nm}$ are the amplitudes of the normal modes of vibration of the plate with sides $2\pi a$, $2\pi b$. He is using the usual plate wave equation 
$$
\frac{\partial^2 y}{\partial t^2}= D \left(\frac{\partial^2 y}{\partial t^x}+\frac{\partial^2 y}{\partial z^2}\right)^2
$$
and expanding the displacment as 
$$
y(x,y,t)= \sum_{n,m} q_{n,m}(t) \cos (mx/a)\cos (ny/b)
$$
A: I think q here denotes a position coordinate. There exists a particular form of potential in which the dynamics of the ball work. It might be that, $q_{mn}$ determines the position of the $m^{\text{th}}$ particle of the ball w.r.t to the $n^{\text{th}}$ particle of the plate. At first, the total potential energy of the $m^{\text{th}}$ particle of the ball due to its interaction with each and every particle of the plate is calculated; that's how the $\sum_{n}$ comes. Next the potential energy of the entire ball is calculated by adding the potential energies of each such particle of the ball; this is how the $\sum_{m}$ comes. However, as both the ball and the plate have continuous mass distribution, the summations should be replaced by integrals.  
${\dot q}_{mn}$ defines the velocity of the $m^{\text{th}}$ particle of the ball w.r.t the $n^{\text{th}}$ particle of the plate.  While calculating the kinetic energy, the summations appear following a similar argument as in the above paragraph and as it seems, they also should be replaced by integrals.
I do not know the exact context of your question though. This is as far as I got from it.
