# Konig's Theorems

In the proof of Konig's Theorems the following identity is used: $$\underline{v_i} = \underline{v_i'} + \underline{v_c}$$ which is the law of transformation of velocity when passing from the inertial frame of reference $$S$$ to the center of mass frame of reference $$S'$$, assuming it doesn't rotate with respect to $$S$$, for otherwise there would be an additional term $$\underline{\omega} \times (\underline{r_i}-\underline{r_c})$$ accounting for rotation. However, my teacher used these theorems despite the fact that the center of mass frame of reference rotated with respect to the inertial one. Nevertheless, he has always reached the correct results, therefore i would like to know if Konig's Theorem has a wider field of validity or if there are certain situations in which it can still be applied without meeting the conditions necessary for its proof.

As always any comment or answer is highly appreciated!

The law relating velocities in both frames is always valid, it's only the expression of $$\vec{v}_c$$ that changes depending on the case.

If $$S'$$ is in pure translation with relation to $$S$$, $$\vec{v}_c=\vec{v}(O'/S)$$, $$O'$$ being the origin of $$S'$$.

If $$S'$$ is in pure uniform rotation with relation to $$S$$, then $$\vec{v}_c$$ is the rotational term you mentioned.

In all other cases, $$\vec{v}_c$$ is the sum of both terms.

When proving König's theorems, you introduce $$S^*$$ which is a specific frame: its origin is the center of mass of the system, and its axes are the same as $$S$$. So $$S^*$$ has no rotation with relation to $$S$$. You don't have to assume there's no rotation, you simply choose a frame that doesn't rotate (usually called the center-of-mass frame).

Small nitpick: $$S^*$$ can have different axes than $$S$$, the only hard requirement is that $$S^*$$'s axes are fixed in $$S$$ (hence no rotation).

• Now it is all much clearer, thank you very much for the answer! May 23 at 10:44