# Calculating the components of angular momentum of a rigid body

You have a rigid body with 1 fixed point in space (the origin).

It's self-explanatory how to get the following equation for the angular momentum:

$\vec L = \sum_n m_n\vec r_n\times\vec v_n$

Where you take the sum of all "$n$" indicates all "$n$" points of mass.
This can be transformed into:

$\vec L = \sum_n m_n(\vec\omega(r_n^2)-(\vec\omega.\vec r_n)\vec r_n)$

Now, out of this, I have to get the following:

For component "$i$" of this equation, you get:
$L_i = \sum_n m_n(r_n^2\omega_i-x_{ni}\sum_j\omega_jx_{nj})$

I understand that for component "$i$", the scalars $m_n$ and $r_n^2$ are the same as for the other components, I also understand that for component "$i$" I need to take the $\omega_i$ component.
But what is meant with everything that follows after that? What does the "$x$" indicate for example? Does the "$j$" indicate the other components?

• – John Alexiou Aug 10 '14 at 15:54
• How is $L_i$ defined in relation to $\vec{L}$? Is this a transformation from the pivot center, to the center of mass? A diagram would help here in order to define the quantities. – John Alexiou Aug 10 '14 at 15:57

While typing this out it clicked for me and I figured it out.
Might as well type the full explanation after typing the question:

Starting from left to right for component "$i$":

• The variables $m_n$ and $r_n^2$ are scalars and so they are the same as for the other components.

• The $\omega_i$ component is self-explanatory as it has the same direction as $L_i$.

• The $x_{ni}$ is to be interpreted as following: "$x$" is a distance (This doesn't indicate your x-component, which had me confused earlier), "$n$" indicates it's for mass-point "$n$" and the $"i"$ indicates it's the $"i"$ component.

• The $\sum_j$ indicates the sum for ALL components (e.g. If you have $x$, $y$ and $z$ components and $i$ is $x$, then $j$ is the sum for $x$, $y$, and $z$.)

• The variables of the summation are to be interpreted in a similar way as earlier.