Palindrom said:
Nobody? I just need a small hint, as I have no intuition yet...
Let me give you a general overview of how the concept of generator is used in Lie algebra.
Look at these differential equations dx/dt = x and dy/dt = y.
I can write these like this \frac {dx^{i}}{dt} = X^{i}(x') where the X^{i} denotes a vector field equal to ( x \frac{d}{dx},y \frac{d}{dy} ) and the x' denotes the variables, here : (x,y).
This field is an operator.
Now suppose we write the solutions of the above differential equaltions as x^{i} = p(t,y), where t is a parameter and y is the solution for t = 0 (Cauchy existence theorem)
If you now expand this solution in a Taylor series with respect to parameter t (so you will get a power series in terms of t and the coefficients are derivatives of solution p with respect to t). You should be able to write down the solution in terms of an exponential that contains the generator X.
Keep in mind that X is defined as the first derivative of the solution p(t,y) with respect to t for t = 0. So you know what the first coefficient is of the t-term in the expansion.
If you are able to do that, you are well on your way.
In extension. If you look at transformations with generator G that leave the differential equation invariant, you will find a nice connection between G (generator of symmetry transformations) and X (generator of the solutions of the differential equation)
regards
marlon
