Angular Momentum

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[edit] In Non-relativistic Classical Mechanics

In classical mechanics the angular momentum of a single particle about a given point $r'$ is given by

$\mathbf{L} = m (\mathbf{r}-\mathbf{r'}) \times \mathbf{v}$

For a system of particles or a continuous object, it is given by

$\mathbf{L} = \sum_{i} m (\mathbf{r_i}-\mathbf{r'}) \times \mathbf{v}, or L = \int_V \rho (\mathbf{r} - \mathbf{r'}) \times \mathbf{v} dv$

In a potential field that is rotationally invariant about a point, the angular momentum of a single particle about that point is always conserved. This holds true for a system of particles or a continuous object by linearity. Note, however, that the angular momentum about other points, (say, the center of mass of the object) is not always conserved. See gravitational stabilization for example

[edit] Of an object rotating about it's center of mass

The angular momentum about an object's center of mass of an object rotating about some point is simply

$\mathbf{L_{a}} = \int_V \rho (\mathbf{r} - \mathbf{r_{a}}) \times \mathbf{v} dv$

Compare this to the angular momentum about any other point of the same motion:

$\mathbf{L_{b}} = \int_V \rho (\mathbf{r} - (\mathbf{r_{a}} - (\mathbf{r_{a}} - \mathbf{r_{b}}))) \times \mathbf{v} dv$ $\mathbf{L_{b}} = \mathbf{L_{a}} - \int_V \rho (\mathbf{r_{a}} - \mathbf{r_{b}}) \times \mathbf{v} dv$

Consider each axis in the remaining integral. The magnitude along these axes is

$\left|{\int_V \rho (\mathbf{r_{CM}} - \mathbf{r_{other}}) \times \mathbf{v} dv}\right| = (\mathbf{r_{CM}} - \mathbf{r_{other}}) \int_V \rho |\mathbf{r} - \mathbf{r_{CM}}| \sin\theta dv$

Since the distribution of mass about the center of mass is by definition balanced, the final integral vanishes, and

$\mathbf{L_{other}} = \mathbf{L_{CM}}$

Here we see that the angular momentum about any axis, of an object rotating about it's center of mass is invariant. Thus a system of objects rotating in this manner may simply have their angular momentums added in an arithmetical fashion.

The center of mass is special. Commonly, the motion of an object is described by the translational velocity of the object $\mathbf{V}$ and the angular velocity of an object $\mathbf{\Omega}$ is defined as the angular velocity of rotation about it's center of mass (ie, with the origin set at the center of mass, $\mathbf{v} = \mathbf{\Omega} \times \mathbf{r}$ ).

[edit] Using the Inertia Tensor

Given an inertia tensor $I$ , the angular momentum about the center of mass is given by

M i = I i k Ω k

(using Einstein notation)

[edit] In Relativistic Mechanics

As is the general pattern in such things, replacing velocity by linear momentum and the cross product by a wedge product yields the relativistic definition