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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 $\vec{\omega_i}\times R_i$?

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  • $\begingroup$ Related answer: physics.stackexchange.com/a/91246/392 $\endgroup$ – ja72 Jan 1 '14 at 17:44
  • $\begingroup$ Can you clarify if $R_i$ is the local to global rotation matrix, or the other way around? $\endgroup$ – ja72 Jan 1 '14 at 18:21
  • $\begingroup$ 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 $\vec{p}_A$. $\endgroup$ – ja72 Jan 1 '14 at 18:38
  • $\begingroup$ is the LHS supposed to be $d\vec{L}/dt$? The first term on the RHS looks like a torque. $\endgroup$ – BMS Jan 1 '14 at 18:39
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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.

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