Suppose I have rigid body of $N$ points in 3D space. I can rotate this body as a whole in that 3D space.
Now suppose I took all coordinates of all points and put them into single $3N$ dimensional vector. Is it a point in "phase" space. And I also can rotate this point in that phase space.
Are these two types of rotations form a single family of things? Can I formulate all rotations of this family in single mathematical manner?