Is this formula for change in angular momentum of a combination of bodies correct?

Question

In my dynamics notes I have written the following: $$\frac{d\vec{p}_A}{dt}= \sum\vec{AC_i}\times m\vec{a_{ci}} + \sum \frac{dR_i}{dt}\left\{{I^{(i)}_{ci}}{\omega}^{(i)}_{ci}\right\}+\sum R\left\{{I^{(i)}_{ci}}\frac{d{\omega}^{(i)}_{ci}}{dt}\right\}$$ where $A$ is a random point, $C_i$ the center of mass of object $i$, $\left\{{I^{(i)}_{ci}}{\omega}^{(i)}_{ci}\right\}$ (an admittedly strange notation for) the resulting vector from $I_{i}\vec{\omega_{ci}}$ if $I_{i}$ is a diagonal matrix and $R_i$ a matrix that projects object $i$ to its principal axes of inertia. I am, however, incapable of finding my derivation of this formula.

Is this formula always correct? And also, -if correct- could $\frac{dR_i}{dt}$ be replace with $\vec{\omega_i}\times R_i$?

Related answer: http://physics.stackexchange.com/a/91246/392 — John Alexiou, Jan 01 '14 at 17:44
Can you clarify if $R_i$ is the local to global rotation matrix, or the other way around? — John Alexiou, Jan 01 '14 at 18:21
I think you need $\sum\vec{AC_i}\times m R_i \vec{a_{ci}}$ in order to transfer the acceleration on the same coordinates as $\vec{p}_A$. — John Alexiou, Jan 01 '14 at 18:38
is the LHS supposed to be $d\vec{L}/dt$? The first term on the RHS looks like a torque. — BMS, Jan 01 '14 at 18:39

score 1 · Answer 1 · edited Apr 13 '17 at 12:40

First off, yes $\frac{dR_i}{dt} = \vec{\omega_i}\times R_i$ if $R_i$ is the local $\rightarrow$ global 3×3 rotation matrix of the i-th body. Meaning that the columns of $R_i$ correspond to the global coordinates of the local $\hat{x}, \hat{y}, \hat{z}$ axes.

I will drop the $i$ subscripts and the $\sum$ and derive the above for one body.

Angular momentum at an arbitrary point A not on the center of mass C is defined as

$$ \vec{p}_A = \vec{p}_C + \vec{r}_{AC} \times \vec{L} = \bar{I} \vec\omega + \vec{r}_{AC} \times m \vec{v}_C $$ where $\vec{L}$ is linear momentum vector, $\vec{\omega} = R\,\vec{\omega}_c$ the angular speed in global coordinates, $\bar{I} = R\,I_c\,R^\top$ the 3×3 inertia tensor in world coordinates, and $\vec{v}_C$ the velocity of the center of mass in world coordinates.

_{(see https://physics.stackexchange.com/a/91246/392 for more details)}

Putting it all together yields

$$ \begin{aligned} \vec{p}_A & = (R\,I_c\,R^\top) R\,\vec{\omega}_c + \vec{r}_{AC} \times m \vec{v}_C \\ & = R\, I_c \vec{\omega}_c + \vec{r}_{AC} \times m \vec{v}_C \end{aligned}$$

Which you can differentiate, but I cannot do unless you explain in details which coordinate system each quantity is defined as. It seems that rotational terms are in local coordinates, and linear terms in world.

Is this formula for change in angular momentum of a combination of bodies correct?

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