I have a doubt, I don't know how to understand generalized velocities, I have seen in the books that they are used like variables. But if they are the derivatives of generalized coordinates, why are they not functions?
I don't know if the generalized velocities are functions, parameters, variables or coordinate axes?
First, we can call 'variable' anything that is not a constant, so generalized velocities will generally be variables, though, depending on the problem, they could also be constants.
On the other hand, generalized velocities are always functions of time, so they are indeed functions. When the function of time is constant, then obviously the corresponding generalized velocity is a constant. However, in general the function of time will not be constant, so the corresponding generalized velocity will be a variable.
In general, we don't call generalized velocities parameters, however, 'parameter' is such a general term in physics that in the right context anything can be called a parameter. For any purpose in Lagrangian Mechanics, it is best to think of time as the only parameter.
Finally, you can view generalized velocities as coordinate axes. It is similar to how you view generalized coordinates as coordinate axes. However, they are axes corresponding to different spaces, i.e. the velocity space and the coordinate space.
