Editing Angular momentum (section)

=== Orbital angular momentum in three dimensions ===
[[File:Torque animation.gif|frame|right|Relationship between [[force]] ('''F'''), [[torque]] ('''τ'''), [[momentum]] ('''p'''), and angular momentum ('''L''') vectors in a rotating system. '''r''' is the [[Position (vector)|position vector]].]]

To completely define orbital angular momentum in [[Three-dimensional space|three dimensions]], it is required to know the rate at which the position vector sweeps out an angle, the direction perpendicular to the instantaneous plane of angular displacement, and the [[mass]] involved, as well as how this mass is distributed in space.<ref>
{{cite book
 |last1 = Watson
 |first1 = W.
 |title = General Physics
 |publisher = Longmans, Green and Co., New York
 |date=1912
 |url=https://books.google.com/books?id=Yb8zAQAAMAAJ
 |page=33
 |via=Google books
}}</ref> By retaining this [[Euclidean vector|vector]] nature of angular momentum, the general nature of the equations is also retained, and can describe any sort of three-dimensional [[Motion (physics)|motion]] about the center of rotation – [[Circular motion|circular]], [[Linear motion|linear]], or otherwise. In [[vector notation]], the orbital angular momentum of a [[point particle]] in motion about the origin can be expressed as:
<math display="block">\mathbf{L} = I\boldsymbol{\omega},</math>
where
* <math>I = r^2m</math> is the [[moment of inertia]] for a [[Point particle|point mass]],
* <math qid=Q161635>\boldsymbol{\omega}=\frac{\mathbf{r}\times\mathbf{v}}{r^2}</math> is the orbital [[angular velocity]] of the particle about the origin,
* <math>\mathbf{r}</math> is the position vector of the particle relative to the origin, and <math>r=\left\vert\mathbf{r}\right\vert</math>,
* <math>\mathbf{v}</math> is the linear velocity of the particle relative to the origin, and
* <math>m</math> is the mass of the particle.

This can be expanded, reduced, and by the rules of [[vector calculus|vector algebra]], rearranged:
<math display="block">\begin{align}
  \mathbf{L} &= \left(r^2m\right)\left(\frac{\mathbf{r}\times\mathbf{v}}{r^2}\right) \\
             &= m\left(\mathbf{r}\times\mathbf{v}\right) \\
             &= \mathbf{r}\times m\mathbf{v} \\
             &= \mathbf{r}\times\mathbf{p},
\end{align}</math>
which is the [[cross product]] of the position vector <math>\mathbf{r}</math> and the linear momentum <math>\mathbf{p} = m\mathbf{v}</math> of the particle. By the definition of the cross product, the <math>\mathbf{L}</math> vector is [[perpendicular]] to both <math>\mathbf{r}</math> and <math>\mathbf{p}</math>. It is directed perpendicular to the plane of angular displacement, as indicated by the [[right-hand rule]] – so that the angular velocity is seen as [[clockwise|counter-clockwise]] from the head of the vector. Conversely, the <math>\mathbf{L}</math> vector defines the [[Plane (geometry)|plane]] in which <math>\mathbf{r}</math> and <math>\mathbf{p}</math> lie.

By defining a [[unit vector]] <math>\mathbf{\hat{u}}</math> perpendicular to the plane of angular displacement, a [[angular frequency|scalar angular speed]] <math>\omega</math> results, where
<math display="block">\omega\mathbf{\hat{u}} = \boldsymbol{\omega},</math> and
<math display="block">\omega = \frac{v_\perp}{r},</math> where <math>v_\perp</math> is the perpendicular component of the motion, as above.

The two-dimensional scalar equations of the previous section can thus be given direction:
<math display="block">\begin{align}
  \mathbf{L} &= I\boldsymbol{\omega}\\
             &= I\omega\mathbf{\hat{u}}\\
             &= \left(r^2m\right)\omega\mathbf{\hat{u}}\\
             &= rmv_\perp \mathbf{\hat{u}}\\
             &= r_\perp mv\mathbf{\hat{u}},
\end{align}</math>
and <math>\mathbf{L} = rmv\mathbf{\hat{u}}</math> for circular motion, where all of the motion is perpendicular to the radius <math>r</math>.

In the [[spherical coordinate system]]  the angular momentum vector expresses as
: <math>\mathbf{L} =
  m \mathbf{r} \times \mathbf{v} = m r^2 \left(\dot\theta\,\hat{\boldsymbol\varphi} - \dot\varphi \sin\theta\,\mathbf{\hat{\boldsymbol\theta}}\right).                            
</math>