Editing Angular momentum (section)

== Solid bodies ==
Angular momentum is also an extremely useful concept for describing rotating rigid bodies such as a [[gyroscope]] or a rocky planet.
For a continuous mass distribution with [[density]] function ''ρ''('''r'''), a differential [[volume element]] ''dV'' with [[position vector]] '''r''' within the mass has a mass element ''dm'' = ''ρ''('''r''')''dV''. Therefore, the [[infinitesimal]] angular momentum of this element is:
<math display="block">d\mathbf{L} = \mathbf{r}\times dm \mathbf{v} = \mathbf{r}\times \rho(\mathbf{r}) dV \mathbf{v} = dV \mathbf{r}\times \rho(\mathbf{r}) \mathbf{v}</math>
and [[volume integral|integrating]] this [[differential (infinitesimal)|differential]] over the volume of the entire mass gives its total angular momentum:
<math display="block">\mathbf{L}=\int_V dV \mathbf{r}\times \rho(\mathbf{r}) \mathbf{v}</math>

In the derivation which follows, integrals similar to this can replace the sums for the case of continuous mass.

=== Collection of particles ===

[[File:Ang mom vector diagram.png|thumb|The angular momentum of the particles ''i'' is the sum of the cross products '''R''' × ''M'''''V''' + Σ'''r'''<sub>''i''</sub> × ''m<sub>i</sub>'''''v'''<sub>''i''</sub>.]]

For a collection of particles in motion about an arbitrary origin, it is informative to develop the equation of angular momentum by resolving their motion into components about their own center of mass and about the origin. Given,
* <math>m_i</math> is the mass of particle <math>i</math>,
* <math>\mathbf{R}_i</math> is the position vector of particle <math>i</math> w.r.t. the origin,
* <math>\mathbf{V}_i</math> is the velocity of particle <math>i</math> w.r.t. the origin,
* <math>\mathbf{R}</math> is the position vector of the center of mass w.r.t. the origin,
* <math>\mathbf{V}</math> is the velocity of the center of mass w.r.t. the origin,
* <math>\mathbf{r}_i</math> is the position vector of particle <math>i</math> w.r.t. the center of mass,
* <math>\mathbf{v}_i</math> is the velocity of particle <math>i</math> w.r.t. the center of mass,

The total mass of the particles is simply their sum,
<math display="block">M=\sum_i m_i.</math>

The position vector of the center of mass is defined by,<ref>
{{cite journal
|last1 = Wilson
|first1 = E. B.
|title = Linear Momentum, Kinetic Energy and Angular Momentum
|journal=The American Mathematical Monthly
|volume=XXII
|publisher = Ginn and Co., Boston, in cooperation with University of Chicago, et al.
|date=1915
|url=https://books.google.com/books?id=nsI0AAAAIAAJ|page= 188, equation (3)
}}</ref>
<math display="block">M\mathbf{R}=\sum_i m_i \mathbf{R}_i.</math>

By inspection,
: <math>\mathbf{R}_i = \mathbf{R} + \mathbf{r}_i</math> and <math>\mathbf{V}_i = \mathbf{V} + \mathbf{v}_i.</math>

The total angular momentum of the collection of particles is the sum of the angular momentum of each particle,
{{Equation box 1 |indent=: |equation =<math>\mathbf{L} = \sum_i \left( \mathbf{R}_i \times m_i \mathbf{V}_i \right)</math> &nbsp;&nbsp;&nbsp; ({{EquationRef|1}}) }}

Expanding <math>\mathbf{R}_i</math>,
: <math>\begin{align}
  \mathbf{L} &= \sum_i \left[\left(\mathbf{R} + \mathbf{r}_i\right) \times m_i\mathbf{V}_i \right] \\
             &= \sum_i \left[ \mathbf{R} \times m_i\mathbf{V}_i + \mathbf{r}_i \times m_i\mathbf{V}_i \right]
\end{align}</math>

Expanding <math>\mathbf{V}_i</math>,
: <math>\begin{align}
  \mathbf{L} &= \sum_i \left[ \mathbf{R} \times m_i\left(\mathbf{V} + \mathbf{v}_i\right) + \mathbf{r}_i \times m_i(\mathbf{V} + \mathbf{v}_i) \right ]\\
             &= \sum_i \left[ \mathbf{R} \times m_i\mathbf{V} + \mathbf{R} \times m_i\mathbf{v}_i + \mathbf{r}_i \times m_i\mathbf{V} + \mathbf{r}_i \times m_i\mathbf{v}_i \right]\\
             &= \sum_i \mathbf{R} \times m_i\mathbf{V} + \sum_i \mathbf{R} \times m_i\mathbf{v}_i + \sum_i \mathbf{r}_i \times m_i\mathbf{V} + \sum_i \mathbf{r}_i \times m_i\mathbf{v}_i
\end{align}</math>

It can be shown that (see sidebar),
{| class="toccolours" style="float:right; margin-left:0.5em; margin-right:0.5em; font-size:84%; background:white; color:black; width:30em; max-width:30%;" cellspacing="5"
| style="text-align:center;" |
'''Prove that''' <math>\sum_i m_i\mathbf{r}_i = \mathbf{0}</math>
<math display="block">\begin{align}
            \mathbf{r}_i &= \mathbf{R}_i - \mathbf{R} \\
         m_i\mathbf{r}_i &= m_i\left(\mathbf{R}_i - \mathbf{R}\right) \\
  \sum_i m_i\mathbf{r}_i &= \sum_i m_i\left(\mathbf{R}_i - \mathbf{R}\right)\\
                         &= \sum_i (m_i\mathbf{R}_i - m_i\mathbf{R})\\
                         &= \sum_i m_i\mathbf{R}_i - \sum_i m_i\mathbf{R}\\
                         &= \sum_i m_i\mathbf{R}_i - \left( \sum_i m_i \right) \mathbf{R}\\
                         &= \sum_i m_i\mathbf{R}_i - M\mathbf{R}
\end{align}</math>
which, by the definition of the center of mass, is <math>\mathbf{0},</math> and similarly for <math display="inline">\sum_i m_i \mathbf{v}_i.</math>
|}
: <math>\sum_i m_i\mathbf{r}_i = \mathbf{0}</math> and <math>\sum_i m_i\mathbf{v}_i = \mathbf{0},</math>
therefore the second and third terms vanish,
: <math>\mathbf{L} = \sum_i \mathbf{R} \times m_i\mathbf{V} + \sum_i \mathbf{r}_i \times m_i\mathbf{v}_i .</math>

The first term can be rearranged,
: <math>\sum_i \mathbf{R} \times m_i\mathbf{V} = \mathbf{R} \times \sum_i m_i\mathbf{V} = \mathbf{R} \times M\mathbf{V},</math>
and total angular momentum for the collection of particles is finally,<ref>
{{cite journal
|last1 = Wilson
|first1 = E. B.
|title = Linear Momentum, Kinetic Energy and Angular Momentum
|journal=The American Mathematical Monthly
|volume=XXII
|publisher = Ginn and Co., Boston, in cooperation with University of Chicago, et al.
|date=1915
|url=https://books.google.com/books?id=nsI0AAAAIAAJ|page=191, Theorem 8
}}</ref>
{{Equation box 1 |indent=: |equation =<math>\mathbf{L} = \mathbf{R} \times M\mathbf{V} + \sum_i \mathbf{r}_i \times m_i\mathbf{v}_i </math> &nbsp;&nbsp;&nbsp; ({{EquationRef|2}}) }}

The first term is the angular momentum of the center of mass relative to the origin. Similar to ''{{section link|#Single particle}}'', below, it is the angular momentum of one particle of mass ''M'' at the center of mass moving with velocity '''V'''. The second term is the angular momentum of the particles moving relative to the center of mass, similar to ''{{section link|#Fixed center of mass}}'', below. The result is general—the motion of the particles is not restricted to rotation or revolution about the origin or center of mass. The particles need not be individual masses, but can be elements of a continuous distribution, such as a solid body.

Rearranging equation ({{EquationNote|2}}) by vector identities, multiplying both terms by "one", and grouping appropriately,
<math display="block">\begin{align}
  \mathbf{L} &= M(\mathbf{R} \times \mathbf{V}) + \sum_i \left[m_i\left(\mathbf{r}_i \times \mathbf{v}_i\right)\right], \\
             &= \frac{R^2}{R^2}M\left(\mathbf{R} \times \mathbf{V}\right) + \sum_i \left[ \frac{r_i^2}{r_i^2}m_i\left(\mathbf{r}_i \times \mathbf{v}_i\right)\right] , \\
             &= R^2M \left( \frac{\mathbf{R} \times \mathbf{V} }{R^2} \right) + \sum_i \left[ r_i^2 m_i \left( \frac{\mathbf{r}_i \times \mathbf{v}_i}{r_i^2} \right) \right] , \\
\end{align}</math>
gives the total angular momentum of the system of particles in terms of [[moment of inertia]] <math>I</math> and [[angular velocity]] <math>\boldsymbol{\omega}</math>,

{{Equation box 1 |indent=: |equation =<math>\mathbf{L} = I_R\boldsymbol{\omega}_R + \sum_i I_i\boldsymbol{\omega}_i.</math> &nbsp;&nbsp;&nbsp; ({{EquationRef|3}}) }}

==== Single particle case ====
In the case of a single particle moving about the arbitrary origin,
<math display="block">\begin{align}
  \mathbf{r}_i &= \mathbf{v}_i = \mathbf{0}, \\
    \mathbf{r} &= \mathbf{R}, \\
    \mathbf{v} &= \mathbf{V}, \\
             m &= M,
\end{align}</math>
<math display="block">\sum_i \mathbf{r}_i \times m_i\mathbf{v}_i = \mathbf{0},</math>
<math display="block">\sum_i I_i\boldsymbol{\omega}_i = \mathbf{0},</math> and equations ({{EquationNote|2}}) and ({{EquationNote|3}}) for total angular momentum reduce to,
<math display="block">\mathbf{L} = \mathbf{R} \times m\mathbf{V} = I_R\boldsymbol{\omega}_R.</math>

==== Case of a fixed center of mass ====
For the case of the center of mass fixed in space with respect to the origin,
<math display="block">\mathbf{V} = \mathbf{0},</math>
<math display="block">\mathbf{R} \times M\mathbf{V} = \mathbf{0},</math>
<math display="block">I_R\boldsymbol{\omega}_R = \mathbf{0},</math> and equations ({{EquationNote|2}}) and ({{EquationNote|3}}) for total angular momentum reduce to,
<math display="block">\mathbf{L} = \sum_i \mathbf{r}_i \times m_i\mathbf{v}_i = \sum_i I_i\boldsymbol{\omega}_i.</math>
{{clear}}