Please provide the simplest example you can think of, of generators of time evolution and generalized coordinates I was reading the Wikipedia article about Noether's theorem and this thing popped out:

Then the resultant perturbation can be written as a linear sum of the
  individual types of perturbations
$$\delta t = \sum_r \epsilon_r T_r \!$$
   $$\delta \mathbf{q} = \sum_r \epsilon_r \mathbf{Q}_r ~$$
where $\epsilon_r$ are infinitesimal parameter coefficients
  corresponding to each:
  
  
*
  
*generator $T_r$ of time evolution, and
  
*generator $Q_r$ of the generalized coordinates.
  

But, when I clic on "generator" it leads me to the article about Lie Groups which is by itself a topic in which I could spend weeks, to say the least.
I had once in my past discrete groups in a lecture and I am more or less conversant with elementary lagrangian mechanics, matrix calculus and elementary non-relativistic quantum mechanics (however not with that SU(?) stuff about rotation invariances yet). Is it possible that someone here provides a very short explanation of what those generators are, and the simplest mathematical example you can think of?
 A: Let us for simplicity address OP's question in the context of point mechanics where $q^i$ are generalized position coordinates on some manifold $M$ [instead of considering field theory with fields $\phi^{\alpha}(x)$]. Let us also for simplicity consider the special case where there are no horizontal/time transformations
$$\delta t ~=~ \sum_a \epsilon_{(a)} X_{(a)} ~=~ 0,$$
but only vertical transformations
$$\delta q^i ~=~ \sum_a \epsilon_{(a)} Y^i_{(a)}, $$
where $\epsilon_{(a)}$ are (infinitesimal) parameters.$^1$
In a nutshell, the generators $Y_{(a)}$ are vector fields
$$ Y_{(a)}~=~ \sum_i Y_{(a)}^i\frac{\partial}{\partial q^i}. $$
The Lie bracket of two vector fields $[Y_{(a)},Y_{(b)}]$  is again a vector field. In certain cases, the generators/vector fields form a Lie algebra.
Example: The generator for a translation in the $a$'th direction of position space $\mathbb{R}^3$ is
$$Y_{(a)}~=~\frac{\partial}{\partial q^a}, $$
with constant vector field components 
$$  Y^i_{(a)}~=~\delta^i_a. $$
In this case, the generators $Y_{(a)}$ commute, so they form an Abelian Lie algebra.
--
$^1$ See also this Phys.SE answer.
