König's theorem (kinetics)
In kinetics, König's theorem or König's decomposition is a mathematical relation derived by Johann Samuel König that assists with the calculations of angular momentum and kinetic energy of bodies and systems of particles. For a system of particles The theorem is divided in two parts. First part of König's theorem The first part expresses the angular momentum of a system as the sum of the angular momentum of the centre of mass and the angular momentum applied to the particles relative to the center of mass. Proof Considering an inertial reference frame with origin O, the angular momentum of the system can be defined as: The position of a single particle can be expressed as: And so we can define the velocity of a single particle: The first equation becomes: But the following terms are equal to zero: So we prove that: where M is the total mass of the system. Second part of König's theorem The second part expresses the kinetic energy of a system of particles in terms of the velocities of the individual particles and the centre of mass. Specifically, it states that the kinetic energy of a system of particles is the sum of the kinetic