How do constraints affect the degree of freedom in particle motion?

[dhruv.tara]

NOTE: This is not a homework problem. So don't move it.

I was reading about degree of freedom and came to know that in general for N particles there are 3N degree of freedom that reduces when we put some constraints on their motion. E.g. For 2 particles confined in XY plane they have only 2 degree of freedom.

But I have a doubt that my source said that not all constraints reduces the degree of freedom, e.g. in the previous case even if I say that the particle lies in only first quadrant of XY plane then too I get 2 degree of freedom. I am not convinced, shouldn't we just have 1 degree of freedom in such a case? (Z=0 and since X and Y are related such that X>0, Y>0)?

Also going on similar basis if I say that I have 2 particles that are confined in a cylinder of radius r and height h then how many degree of freedoms do I have? Do I have 4 degree of freedom or 5?

Thanks in advance.

 

[Feldoh]

Also going on similar basis if I say that I have 2 particles that are confined in a cylinder of radius r and height h then how many degree of freedoms do I have? Do I have 4 degree of freedom or 5?

Because x and y are related it counts as 1 DOF per particle (unless the particles are somehow related). Or if you like we could say that the particles each move in a angular direction(theta, phi, etc. And of course 1 DOF per particle for moving up and down on the cylinder.

But I have a doubt that my source said that not all constraints reduces the degree of freedom, e.g. in the previous case even if I say that the particle lies in only first quadrant of XY plane then too I get 2 degree of freedom. I am not convinced, shouldn't we just have 1 degree of freedom in such a case? (Z=0 and since X and Y are related such that X>0, Y>0)?

There may be constraints but as I understand it that since x and y are not related to each other the DOF do not reduce since we still have both x movement and y, movement independently.