Theorems of König (mechanics)

In mechanics, the sentences of König (after Johann Samuel König ) are two related statements about the angular momentum ( 1st sentence of König ) or the kinetic energy ( 2nd sentence of König ) of a system of mass points which these two quantities express it in a way that is physically easy to interpret.

1. King's theorem

König's 1st theorem reads: The angular momentum of the system with respect to the origin is the sum of the angular momentum of the center of gravity around the origin, provided with the total mass of the system and the intrinsic angular momentum of the system with respect to the center of gravity: ${\displaystyle {\vec {L}}}$ ${\displaystyle M}$ ${\displaystyle {\vec {J}}}$

${\displaystyle {\vec {L}}=M{\vec {r}}_{\mathrm {S} }\times {\vec {v}}_{\mathrm {S} }+{\vec {J}}}$

It applies

${\displaystyle {\vec {L}}=\sum m_{i}{\vec {r}}_{i}\times {\vec {v}}_{i}=\sum m_{i}{\vec {r}}_{i}\times ({\vec {v}}_{\mathrm {S} }+{\vec {v}}_{\mathrm {rel} ,i})=M{\vec {r}}_{\mathrm {S} }\times {\vec {v}}_{\mathrm {S} }+\sum m_{i}{\vec {r}}_{i}\times {\vec {v}}_{\mathrm {rel} ,i}}$

with the definition of the focus ${\displaystyle {\vec {r}}_{\mathrm {S} }}$

${\displaystyle {\vec {r}}_{\mathrm {S} }={\frac {\sum m_{i}{\vec {r}}_{i}}{M}}}$

and the center of gravity speed . In particular, it is irrelevant for the intrinsic angular momentum whether it relates to the origin or the center of gravity, because it is: ${\displaystyle {\vec {v}}_{\mathrm {S} }={\dot {\vec {r}}}_{\mathrm {S} }}$

${\displaystyle {\vec {J}}=\sum m_{i}{\vec {r}}_{i}\times {\vec {v}}_{\mathrm {rel} ,i}=\sum m_{i}({\vec {r}}_{i}-{\vec {r}}_{\mathrm {S} })\times {\vec {v}}_{\mathrm {rel} ,i}+\sum m_{i}{\vec {r}}_{S}\times {\vec {v}}_{\mathrm {rel} ,i}}$

Since the sum of the relative momentum vanishes in the center of gravity system, the second term is identical to zero.

2nd theorem of König

König's 2nd theorem reads: The kinetic energy of a system is the sum of the kinetic energy of the movement of the center of gravity and its internal kinetic energy: ${\displaystyle E}$

${\displaystyle E={\frac {1}{2}}M{\vec {v}}_{\mathrm {S} }^{2}+{\frac {1}{2}}\sum m_{i}{\vec {v}}_{\mathrm {rel} ,i}^{2}}$

The following applies:

${\displaystyle E={\frac {1}{2}}\sum m_{i}{\vec {v}}_{i}^{2}={\frac {1}{2}}\sum m_{i}({\vec {v}}_{\mathrm {S} }+{\vec {v}}_{\mathrm {rel} ,i})^{2}={\frac {1}{2}}\sum m_{i}({\vec {v}}_{\mathrm {S} }^{2}+2{\vec {v}}_{\mathrm {S} }\cdot {\vec {v}}_{\mathrm {rel} ,i}+{\vec {v}}_{\mathrm {rel} ,i}^{2})={\frac {1}{2}}M{\vec {v}}_{\mathrm {S} }^{2}+\sum m_{i}{\vec {v}}_{\mathrm {S} }\cdot {\vec {v}}_{\mathrm {rel} ,i}+{\frac {1}{2}}\sum m_{i}{\vec {v}}_{\mathrm {rel} ,i}^{2}}$

Due to the same argument as in König's 1st theorem, the second summand disappears.

Applications

König's two theorems hold true whether the system is deformable or not. They are often used in the particularly important rigid body case .

literature

• Samuel König: De universali principio aequilibrii & motus, in vi viva reperto, deque nexu inter vim vivam & actionem, utriusque minimo, dissertatio . 1751 (Latin).