König's theorem (kinetics)

For other uses, see König's theorem (disambiguation).

In kinetics, König's theorem or König's decomposition is a mathematical relation derived by Johann Samuel König that assists with the calculation of kinetic energy of bodies and systems of particles.

For a system of particles

The theorem expresses the kinetic energy of a system of particles in terms of the velocities of the individual particles.

Specifically, it states that the kinetic energy of a system of particles is the sum of the kinetic energy associated to the movement of the center of mass and the kinetic energy associated to the movement of the particles relative to the center of mass.

For a rigid body

The theorem can also be applied to rigid bodies, stating that the kinetic energy T of a rigid body, as viewed by an observer fixed in some inertial reference frame N, can be written as:

$^{N}T={\frac {1}{2}}m\cdot {^{N}\mathbf {\bar {v}} }\cdot {^{N}\mathbf {\bar {v}} }+{\frac {1}{2}}{^{N}\!\mathbf {\bar {H}} }\cdot ^{N}{\!\!\mathbf {\omega } }^{R}$

where ${m}$ is the mass of the rigid body; ${^{N}\mathbf {\bar {v}} }$ is the velocity of the center of mass of the rigid body, as viewed by an observer fixed in an inertial frame N; ${^{N}\!\mathbf {\bar {H}} }$ is the angular momentum of the rigid body about the center of mass, also taken in the inertial frame N; and $^{N}{\!\!\mathbf {\omega } }^{R}$ is the angular velocity of the rigid body R relative to the inertial frame N.

References

• Samuel König (Sam. Koenigio): De universali principio æquilibrii & motus, in vi viva reperto, deque nexu inter vim vivam & actionem, utriusque minimo, dissertatio, Nova acta eruditorum (1751) 125-135, 162-176 (Archived).
• Paul A. Tipler and Gene Mosca (2003), Physics for Scientists and Engineers (Paper) : Volume 1A: Mechanics (Physics for Scientists and Engineers), W. H. Freeman Ed., ISBN 0-7167-0900-7