König's theorem (kinetics)

As a set of king ( after Johann Samuel King ) is called in mechanics two related statements about the angular momentum ( 1 set of king ) and the kinetic energy ( second set of king ) of a system of mass points that these two quantities expressed on a physically easy to interpret nature.

The concept of the focus system

In two statements one makes a specific reference system advantage: the focus system, which we denote by (R *). Contrast, is denoted by (R ), the coordinate system of our choice, we start from. (R) may be an inertial or not.

Definition

By definition, (R * ) is the coordinate system that can be seen from (R ) by a translation, so that the total momentum of the system under consideration of mass points disappears in (R *). This is the general definition, which remain valid in the relativistic case.

In Newtonian mechanics, the total momentum can be known easily expressed using the motion of the center of gravity G: with the total mass. In the center of mass system is therefore, and whereas it is the other usual definition of (R * ) (R *) is the coordinate system in which the center of gravity G and the rest is clear from (R ) by a translational movement.

Properties of ( R *)

Note: (R *) is an inertial frame if and only if an (R ) is an inertial system.

Is the angular momentum with respect to the point O, and the angular momentum with respect to the point O '. Then quite general:. However, since by definition in (R *), the angular momentum of the system in (R *) is independent of the reference point.   is also called intrinsic angular momentum or internal angular momentum of the system.

On the other hand is considered, the general formula for the total angular momentum, however, ( the addition of the speed ), and therefore: .

But according to the definition of gravity: and since, we obtain the following fundamental property:

.

Finally, in (R *) defining the internal kinetic energy of the system.

1 set of king

Statement: With the above notation applies:

Physical interpretation: The angular momentum of the system with respect to a point O is the sum of two terms:

• Of the angular momentum of the center of gravity G, provided with the total mass M of the system;
• And intrinsic angular momentum of the system, is identical to that in (R) with respect to the point G calculated angular momentum of the system.

Proof: From the general expression for the angular momentum with respect to the point O in the reference system (R ) and from (addition of velocities ) follows:. Now, by definition, and the center of gravity, so the following (1).

Second set of king

Statement: With the above notation applies:

Physical interpretation: The kinetic energy of the system is the sum of two terms:

• The kinetic energy of the center of gravity G, provided with the total mass M of the system;
• And the internal kinetic energy of the system.

Proof: As stated:. When employing the general expression for the kinetic energy of a system, one obtains:

, Of the first term of the right side is no more than, and the total mass and by the definition of (R * ), ie the following (2).

Applications

The two sets of king apply, regardless of whether the system is deformed or not. They are often used in the particularly important case of the rigid body.

• Theoretical Mechanics