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
Angular velocity which shows a connection with the Lie group of rotations. We prove that angular velocity tensor is skew symmetric, i.e. formula_63 satisfies formula_64. A rotation matrix "A" is orthogonal, inverse to its transpose, so we have formula_65. For formula_66 a frame matrix, taking the time derivative of the equation gives: Applying the formula formula_68, Thus, "W" is the negative of its transpose, which implies it is skew symmetric. At any instant formula_70, the angular velocity tensor represents a linear map between the position vector formula_71 and the velocity vectors formula_72 of a point on a rigid body rotating around the