Degrees of freedom an constraints

[aaaa202]

I'm not quite sure I get the idea of a degree of freedom for a system. First of all: Is there freedom in characterizing the DOF for a system - i.e. will specifying the DOF for a system relative to any coordinate system always be the same?
Next let me do an example: If we have 2 particles free to rotate about any axis, what is the total DOF for that system? Is that 12 or 6? Because I normally see 6, but isn't that just because you see the particle as a point? On the other hand it perhaps doesn't make sense to say that a single particle can rotate.
Next let's imagine that we put a constraint on the system saying that the distance between the two particles must stay fixed. I have then been told that the total DOF are 5. But how do I realize that? And does this number account for rotational DOF?

 

[hotvette]

One way to look at this is to ask the question: how many numeric values does it take to completely describe the state of the system? You first have to have some assumptions about valid states: translation, rotation, time, etc.

In the case of 5 DOF, I am guessing that position is the only valid state (i.e. rotations aren't allowed). In that case, the first particle is described by 3 spatial coordinates. The 2nd particle can be anywhere on a fixed sphere around that particle. In spherical coordinates, r is fixed but theta and phi can be anything, thus 2 extra DOF for a total of 5 DOF.