Total Angular Momentum
Work in Progress by Fatima Jamil
Total angular momentum can be expressed as 
. This page explains the breakdown of total angular momentum in these 2 components to help understand the difference between rotational angular momentum and translational angular momentum.
The Main Idea
It is conveniant to break apart total angular momentum for a multiparticle system into rotational angular momentum and translational angular momentum. The translational angular momentum is associated with a rotation of the center of mass about some point A. This differs for different choices of the location of point A. The rotational angular momentum is associated with a rotation about the center of mass. The rotational angular momentum is independent of the location of the point A and the motion of the center of mass.
A Mathematical Model
It is often convenient to consider the angular momentum of a collection of particles about their center of mass, because this simplifies the mathematics considerably. The angular momentum of a collection of particles is the sum of the angular momenta of each particle:
[math]\displaystyle{ \mathbf{L}=\sum_i \mathbf{R}_i\times m_i \mathbf{V}_i }[/math] where [math]\displaystyle{ R_i }[/math] is the distance of particle i from the reference point, [math]\displaystyle{ m_i }[/math] is its mass, and [math]\displaystyle{ V_i }[/math] is its velocity. The center of mass is defined by:
[math]\displaystyle{ \mathbf{R}=\frac{1}{M}\sum_i m_i \mathbf{R}_i }[/math] where [math]\displaystyle{ M }[/math] is the total mass of all the particles.
If we define [math]\displaystyle{ \mathbf{r}_i }[/math] as the displacement of particle i from the center of mass, and [math]\displaystyle{ \mathbf{v}_i }[/math] as the velocity of particle i with respect to the center of mass, then we have
[math]\displaystyle{ \mathbf{R}_i=\mathbf{R}+\mathbf{r}_i\, }[/math] and [math]\displaystyle{ \mathbf{V}_i=\mathbf{V}+\mathbf{v}_i\, }[/math] In this case, the total angular momentum is:
[math]\displaystyle{ \mathbf{L}=\sum_i (\mathbf{R}+\mathbf{r}_i)\times m_i (\mathbf{V}+\mathbf{v}_i) = \left(\mathbf{R}\times M\mathbf{V}\right) + \left(\sum_i \mathbf{r}_i\times m_i \mathbf{v}_i\right) }[/math]
A Computational Model
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