Quote by nikolafmf
If u=(x,y,vx,vy), then du/dt = (vx, vy, GMx/(x^2+y^2)^(3/2), GMx/(x^2+y^2)^(3/2)), right?

That is correct (well, the last component should be proportional with y and not x, but I assume that is just a typo).
Then what function of u is it? I see that it is a function of the components of u, but not of u itself. And, t doesn't even appear here (at least not explicitely).

The vector u is just a convenient notation for the complete state  a notion used by the RK scheme you mentioned part of. If you are unsure about what a vector is you probably want to refer to a textbook for some background (see [1] for a brief overview). You are free to translate each vector equation, like k
_{1} = f(u), to the corresponding set of 4 equations, like
k
_{1}[0] = f(u
_{n})[0] = (vx
_{n}, vy
_{n}, GMx/(x
_{n}^{2}+y
_{n}^{2})
^{(3/2)}, GMy/(x
_{n}^{2}+y
_{n}^{2})
^{(3/2)})[0] = vx
_{n},
k
_{1}[1] = f(u
_{n})[1] = ... = vy
_{n},
k
_{1}[2] = f(u
_{n})[1] = ... = GMx/(x
_{n}^{2}+y
_{n}^{2})
^{(3/2)}, and
k
_{1}[3] = f(u
_{n})[3] = ... = GMy/(x
_{n}^{2}+y
_{n}^{2})
^{(3/2)}
where I have used the codelike notation "vector[i]" to mean "the ith component of vector" for i = 0, 1, 2, 3. So, for each of k
_{1} to k
_{4} you can write up such 4 scalar expressions.
You are correct that time t does not explicitly appear in the field, that is, f(u, t) = f(u). This is quite common in dynamical systems and a system with this characteristic is called an autonomous system (see for instance [2]).
[1]
http://en.wikipedia.org/wiki/Euclidean_vector
[2]
http://en.wikipedia.org/wiki/Autonom...m_(mathematics)