I would like to know why in the context of vibrating systems, we define degrees of freedom in terms of number of independent coordinates (by coordinates I mean the numbers which specify the components of vectors) required to specify the motion instead of the configuration (configuration refers to just the positions and not velocities). Are these definitions equivalent because for motion, we are always concerned with the rate of change of each coordinate (describing the position or configuration) and thus motion also ends up involving same number of coordinates to describe the motion?
The book I am reading also mentions a line saying
On occasion, when internal forces are carried as unknowns, the number of equations exceeds the number of degrees of freedom by the number of unknown forces.
I am not able to see where would such a need arise; an example of this would be greatly appreciated.