Angular Momentum Vector

Like linear momentum, angular momentum is fundamentally a vector in $\mathbb{R}^3$ . The definition of the previous section suffices when the direction does not change, in which case we can focus only on its magnitude $L=I\omega$ .

More generally, let $\underline{x}_m\in\mathbb{R}^3$ denote the 3-space coordinates of a point-mass $m$ , and let $\underline{v}_m=\dot{\underline{x}}_m$ denote its velocity in $\mathbb{R}^3$ . Then the instantaneous angular momentum vector of the mass relative to the origin (not necessarily rotating about a fixed axis) is given by

$$\underline{L}\isdefs m\, \underline{x}_m \times \underline{v}_m \eqsp m\, \underline{x}_m \times (\underline{\omega}\times\underline{x}_m) \protect$$ (B.19)

where $\times$ denotes the vector cross product, discussed in §B.4.12 above. The identity $\underline{v}_m=\underline{\omega}\times\underline{x}_m$ was discussed at Eq.(B.17).

For the special case in which $\underline{v}_m$ is orthogonal to $\underline{x}_m$ , as in Fig.B.4, we have that $x_m\times\underline{v}_m$ points, by the right-hand rule, in the direction of the angular velocity vector $\underline {\omega }$ (up out of the page), which is satisfying. Furthermore, its magnitude is given by

$$\left\Vert\,\underline{L}\,\right\Vert \eqsp m\left\Vert\,\underline{x}_m\times\underline{v}_m\,\right\Vert \eqsp m\left\Vert\,\underline{x}_m\,\right\Vert\left\Vert\,\underline{v}_m\,\right\Vert \eqsp mRv \eqsp mR^2\omega \eqsp I\omega$$

which agrees with the scalar case.

In the more general case of an arbitrary mass velocity vector $\underline{v}_m$ , we know from §B.4.12 that the magnitude of $\underline{x}_m\times\underline{v}_m$ equals the product of the distance from the axis of rotation to the mass, i.e., $\vert\vert\,\underline{x}_m\,\vert\vert$ , times the length of the component of $\underline{v}_m$ that is orthogonal to $\underline {x}$ , i.e., $\vert\vert\,\underline{v}_m\,\vert\vert \sin(\theta)$ , as needed.

It can be shown that vector angular momentum, as defined, is conserved.^B.22 For example, in an orbit, such as that of the moon around the earth, or that of Halley's comet around the sun, the orbiting object speeds up as it comes closer to the object it is orbiting. (See Kepler's laws of planetary motion.) Similarly, a spinning ice-skater spins faster when pulling in arms to reduce the moment of inertia about the spin axis. The conservation of angular momentum can be shown to result from the principle of least action and the isotrophy of space [272, p. 18].