Koenig theorem (mechanics)

Koenig's theorem allows us to express the total kinetic energy of a mechanical system through the energy of motion of the center of mass and the energy of motion relative to the center of mass. It was formulated and proved by I. S. Koenig in 1751. ^[1]

Content

1 Formulation
2 Conclusion
3 See also
4 notes
5 Literature

Wording

The kinetic energy of a mechanical system is the energy of motion of the center of mass plus the energy of motion relative to the center of mass:

{T\;=\;T_{0}+T_{r}}\;,

{\ displaystyle {T \; = \; T_ {0} + T_ {r}} \ ;,}

Where $T$ ${\ displaystyle T}$ Is the total kinetic energy of the system, $T_{0}$ ${\ displaystyle T_ {0}}$ - kinetic energy of the motion of the center of mass, $T_{r}$ ${\ displaystyle T_ {r}}$ Is the relative kinetic energy of the system ^[2] .

In other words, the total kinetic energy of a body or system of bodies in complex motion is equal to the sum of the energy of the system in translational motion and the energy of the system in its motion relative to the center of mass.

More precise wording ^[3] :

The kinetic energy of the system of material points is equal to the sum of the kinetic energy of the entire mass of the system, mentally concentrated in its center of mass and moving with it, and the kinetic energy of the same system in its relative motion with respect to the translationally moving coordinate system with the origin at the center of mass.

Conclusion

We give a proof of Koenig's theorem for the case when the masses of bodies that form a mechanical system $S$ ${\ displaystyle S}$ are distributed continuously ^[4] .

Find the relative kinetic energy $T_{r}$ ${\ displaystyle T_ {r}}$ the system $S$ ${\ displaystyle S}$ treating it as kinetic energy calculated relative to a moving coordinate system . Let be ${\vec {\rho }}$ ${\ displaystyle {\ vec {\ rho}}}$ Is the radius vector of the considered point of the system $S$ ${\ displaystyle S}$ in a moving coordinate system. Then ^[5] :

T_{r}\;=\;{\frac {1}{2}}\int {\frac {{\rm {d}}{\vec {\rho }}}{{\rm {d}}t}}\cdot {\frac {{\rm {d}}{\vec {\rho }}}{{\rm {d}}t}}\,{\rm {d}}m\;,

{\ displaystyle T_ {r} \; = \; {\ frac {1} {2}} \ int {\ frac {{\ rm {d}} {\ vec {\ rho}}} {{\ rm {d }} t}} \ cdot {\ frac {{\ rm {d}} {\ vec {\ rho}}} {{\ rm {d}} t}} \, {\ rm {d}} m \; ,}

where the dot denotes the scalar product , and integration is carried out over the area of space occupied by the system at the current time.

If ${\vec {r}}_{0}$ ${\ displaystyle {\ vec {r}} _ {0}}$ Is the radius vector of the origin of the coordinates of the mobile system, and ${\vec {r}}$ ${\ displaystyle {\ vec {r}}}$ Is the radius vector of the considered point of the system $S$ ${\ displaystyle S}$ in the original coordinate system, the relation is true:

{\vec {r}}\;=\;{\vec {r}}_{0}+{\vec {\rho }}\;.

{\ displaystyle {\ vec {r}} \; = \; {\ vec {r}} _ {0} + {\ vec {\ rho}} \ ;.}

We calculate the total kinetic energy of the system in the case when the origin of the moving system is placed at its center of mass. Taking into account the previous relation, we have:

T\;=\;{\frac {1}{2}}\int {\frac {{\rm {d}}{\vec {r}}}{{\rm {d}}t}}\cdot {\frac {{\rm {d}}{\vec {r}}}{{\rm {d}}t}}\,{\rm {d}}m\;=\;{\frac {1}{2}}\int \left({\frac {{\rm {d}}{\vec {r}}_{o}}{{\rm {d}}t}}+{\frac {{\rm {d}}{\vec {\rho }}}{{\rm {d}}t}}\right)\cdot \left({\frac {{\rm {d}}{\vec {r}}_{o}}{{\rm {d}}t}}+{\frac {{\rm {d}}{\vec {\rho }}}{{\rm {d}}t}}\right)\,{\rm {d}}m\;.

{\ displaystyle T \; = \; {\ frac {1} {2}} \ int {\ frac {{\ rm {d}} {\ vec {r}}} {{\ rm {d}} t} } \ cdot {\ frac {{\ rm {d}} {\ vec {r}}} {{\ rm {d}} t}} \, {\ rm {d}} m \; = \; {\ frac {1} {2}} \ int \ left ({\ frac {{\ rm {d}} {\ vec {r}} _ {o}} {{\ rm {d}} t}} + {\ frac {{\ rm {d}} {\ vec {\ rho}}} {{\ rm {d}} t}} \ right) \ cdot \ left ({\ frac {{\ rm {d}} {\ vec {r}} _ {o}} {{\ rm {d}} t}} + {\ frac {{\ rm {d}} {\ vec {\ rho}}} {{\ rm {d}} t}} \ right) \, {\ rm {d}} m \ ;.}

Given that the radius vector ${\vec {r}}_{0}$ ${\ displaystyle {\ vec {r}} _ {0}}$ same for everyone ${\rm {d}}m$ ${\ displaystyle {\ rm {d}} m}$ , you can open the brackets ${\frac {{\rm {d}}{\vec {r}}_{0}}{{\rm {d}}t}}$ ${\ displaystyle {\ frac {{\ rm {d}} {\ vec {r}} _ {0}} {{\ rm {d}} t}}}$ per integral sign:

T\;=\;{\frac {1}{2}}{\frac {{\rm {d}}{\vec {r}}_{0}}{{\rm {d}}t}}\cdot {\frac {{\rm {d}}{\vec {r}}_{0}}{{\rm {d}}t}}\int \,{\rm {d}}m\,\,+\,\,{\frac {{\rm {d}}{\vec {r}}_{0}}{{\rm {d}}t}}\cdot \int {\frac {{\rm {d}}{\vec {\rho }}}{{\rm {d}}t}}\,{\rm {d}}m\,\,+\,\,{\frac {1}{2}}\int {\frac {{\rm {d}}{\vec {\rho }}}{{\rm {d}}t}}\cdot {\frac {{\rm {d}}{\vec {\rho }}}{{\rm {d}}t}}\,{\rm {d}}m\;.

{\ displaystyle T \; = \; {\ frac {1} {2}} {\ frac {{\ rm {d}} {\ vec {r}} _ {0}} {{\ rm {d}} t}} \ cdot {\ frac {{\ rm {d}} {\ vec {r}} _ {0}} {{\ rm {d}} t}} \ int \, {\ rm {d}} m \, \, + \, \, {\ frac {{\ rm {d}} {\ vec {r}} _ {0}} {{\ rm {d}} t}} \ cdot \ int {\ frac {{\ rm {d}} {\ vec {\ rho}}} {{\ rm {d}} t}} \, {\ rm {d}} m \, \, + \, \, {\ frac {1} {2}} \ int {\ frac {{\ rm {d}} {\ vec {\ rho}}} {{\ rm {d}} t}} \ cdot {\ frac {{\ rm {d}} {\ vec {\ rho}}} {{\ rm {d}} t}} \, {\ rm {d}} m \ ;.}

The first term on the right-hand side of this formula (which coincides with the kinetic energy of the material point, which is placed at the origin of the moving system and has a mass equal to the mass of the mechanical system) can be interpreted ^[2] as the kinetic energy of the center of mass.

The second term is equal to zero, since the second factor in it is obtained by differentiating with respect to time the product of the radius vector of the center of mass and the mass of the system ^[6] , but the mentioned radius vector (and with it the whole product) is equal to zero:

\int {\vec {\rho }}\,\,{\rm {d}}m=0\;,

{\ displaystyle \ int {\ vec {\ rho}} \, \, {\ rm {d}} m = 0 \ ;,}

since the origin of the coordinates of the moving system is (by assumption) in the center of mass.

The third term, as already shown, is $T_{r}$ ${\ displaystyle T_ {r}}$ i.e. the relative kinetic energy of the system $S$ ${\ displaystyle S}$ .

Notes

↑ Gernet, 1987 , p. 258.
↑ ¹ ² Zhuravlev, 2001 , p. 72.
↑ Sivukhin D.V. General course of physics. - M .: Fizmatlit , 2005. - T. I. Mechanics. - S. 137-138. - 560 s. - ISBN 5-9221-0225-7 .
↑ Zhuravlev, 2001 , p. 71-72.
↑ Zhuravlev, 2001 , p. 71.
↑ Zhuravlev, 2001 , p. 66.

Literature

Gernet M.M. Course in Theoretical Mechanics. 5th ed. - M .: Higher school, 1987 .-- 344 p.
Zhuravlev V.F. Fundamentals of Theoretical Mechanics. 2nd ed. - M .: Fizmatlit, 2001 .-- 320 p. - ISBN 5-94052-041-3 .

[_0b0b8fde306efb01-1] Gernet, 1987 , p. 258.

[_5bb157eef27335fd-2] ¹ ² Zhuravlev, 2001 , p. 72.

[3] Sivukhin D.V. General course of physics. - M .: Fizmatlit , 2005. - T. I. Mechanics. - S. 137-138. - 560 s. - ISBN 5-9221-0225-7 .

[_a7a428443d94c1db-4] Zhuravlev, 2001 , p. 71-72.

[_5bb157eef27335fe-5] Zhuravlev, 2001 , p. 71.

[_5bb156eef273342a-6] Zhuravlev, 2001 , p. 66.