Consider a system of two particles of masses $m_1$ and $m_2$ moving in a plane. Let the respective position vectors be $\mathbf{r_1}$ and $\mathbf{r_2}$. The particles are attached at the end of a massless rigid rod of some length $l$. Since this is a holonomic constraint, $$(\mathbf{r_1-r_2})^2-l^2=0$$ The system should have $4-1=3$ generalized independent coordinates. I am apparently unable to figure out those.
Since there is only one constraint, fixing any three of the four cartesian coordinates (coordinates of both the particles) should automatically, I think, fix the remaining one. But given $(x_1,y_1)$ and $x_2$, $y_2$ can have two values that satisfy the constraint. So is $y_2$ independent in some way?
About the generalized coordinates, let $\mathbf{R}$ (2 d.o.f) be the C.O.M. of the system. For third d.o.f, take the angle $\mathbf{r_1-r_2}$ makes with the positive x-axis. But if only the magnitude of that angle is considered, there will be two possible configurations of the rod for every possible value of that angle (above the axis and below the axis). To tackle this, one way is to assign a sign to the angle, + above the axis from $0$ to $\pi/2$ (counter-clockwise) and (-) below the axis from $0$ to $-\pi/2$ (clockwise). Would this be a legit way?
