Center of mass rigid body, the gravity forces will not cause the body to rotate, which means the weight of the body can be considered to be concentrated at the center of mass. The linear and angular momentum of a collection of particles can be simplified by measuring the position and velocity of the particles relative to the center of mass. Let the system of particles "P", "i"=1...,"n" of masses "m" be located at the coordinates r with velocities v. Select a reference point R and compute the relative position and velocity vectors, The total linear and angular momentum vectors relative to the
Cross product The cross product with n therefore describes the infinitesimal generator of the rotations about n. These infinitesimal generators form the Lie algebra so(3) of the rotation group SO(3), and we obtain the result that the Lie algebra R with cross product is isomorphic to the Lie algebra so(3). The vector cross product also can be expressed as the product of a skew-symmetric matrix and a vector: where superscript refers to the transpose operation, and [a] is defined by: The columns [a] of the skew-symmetric matrix for a vector a can be also obtained by calculating the cross product with unit