# Independent coordinates of a rigid body

This is a quote from Classical mechanics by Goldstein:

"To fix a point in a rigid body, it is not necessary to specify its distances to all other points in the body ; we need only state the distances to any three other non-collinear points in the body. Thus, once the positions of three of the particles of the rigid body are determined. The constraints fix the positions of all remaining particles. The number of degrees of freedom therefore cannot be more than nine."

I'm having trouble unpacking this. If the body was a two dimensional plane, then the position vector of some particle in an external reference frame could be described by the difference vector between two points of that plane. Are we using the same concept here?

The degrees of freedom are connected to this, but it seems easier to understand. The distances between the three points are fixed. Thus, they can only rotate about themselves in the $$x,y,z$$ directions. The constraints ( distances between the points are constant ) further reduce these to $$3$$ DOF's.

3D case

the Rigid body has 6 generalized coordinates 3 translation and 3 rotation

you have three non colinear points $$~\bf R_1~,\bf R_2~,\bf R_3$$ on the rigid body, each point has three degrees of freedom $$~x~,y~,z$$

you obtain three constraint equations

$$\parallel \bf{R}_1-\bf R_2\parallel=\text{const.}\\ \parallel \bf R_1-\bf R_3\parallel=\text{const.}\\ \parallel \bf R_3-\bf R_2\parallel=\text{const.}$$

so that ,the number of the generalized coordinates is $$~3\cdot 3-3=6~$$

the rigid body can rotate about the axes

$$\bf R_1\times \bf R_2\\ \bf R_1\times \bf R_3\\ \bf R_3\times \bf R_2$$

2D case

the Rigid body has 3 generalized coordinates 2 translation and 1 rotation

you have two non colinear points $$~\bf R_1~,\bf R_2$$ each point has two degrees of freedom $$~x~,y$$

and the constraint equation $$~\parallel \bf{R}_1-\bf R_2\parallel=\text{const.}$$

thus the number of the generalized coordinates is $$~2\cdot 2-1=3~$$

the body can rotate about the axis $$~\bf R_1\times \bf R_2~$$