Editing Moment of inertia (section)

=== Definition ===

For a rigid object of <math>N</math> point masses <math>m_{k}</math>, the moment of inertia [[tensor]] is given by
<math display="block">
\mathbf{I} = \begin{bmatrix}
I_{11} & I_{12} & I_{13} \\
I_{21} & I_{22} & I_{23} \\
I_{31} & I_{32} & I_{33}
\end{bmatrix}.
</math>

Its components are defined as
<math display="block">I_{ij} \ \stackrel{\mathrm{def}}{=}\  \sum_{k=1}^{N} m_{k}\left(\left\|\mathbf{r}_k\right\|^{2}\delta_{ij} - x_{i}^{(k)}x_{j}^{(k)}\right)</math>
where
* <math>i</math>, <math>j</math> is equal to 1, 2 or 3 for <math>x</math>, <math>y</math>, and <math>z</math>, respectively,
* <math>\mathbf{r}_k = \left(x_1^{(k)}, x_2^{(k)}, x_3^{(k)}\right)</math> is the vector to the point mass <math>m_k</math> from the point about which the tensor is calculated and
* <math>\delta_{ij}</math> is the [[Kronecker delta]].

Note that, by the definition, <math>\mathbf{I}</math> is a [[symmetric tensor]].

The diagonal elements are more succinctly written as
<math display="block">\begin{align}
  I_{xx} \ &\stackrel{\mathrm{def}}{=}\  \sum_{k=1}^{N} m_{k} \left(y_{k}^{2} + z_{k}^{2}\right), \\
  I_{yy} \ &\stackrel{\mathrm{def}}{=}\  \sum_{k=1}^{N} m_{k} \left(x_{k}^{2} + z_{k}^{2}\right), \\
  I_{zz} \ &\stackrel{\mathrm{def}}{=}\  \sum_{k=1}^{N} m_{k} \left(x_{k}^{2} + y_{k}^{2}\right),
\end{align}</math>
while the off-diagonal elements, also called the '''{{Interlanguage link|Produit d'inertie|fr|lt=products of inertia}}''', are
<math display="block">\begin{align}
  I_{xy} = I_{yx} \ &\stackrel{\mathrm{def}}{=}\  -\sum_{k=1}^{N} m_{k} x_{k} y_{k}, \\
  I_{xz} = I_{zx} \ &\stackrel{\mathrm{def}}{=}\  -\sum_{k=1}^{N} m_{k} x_{k} z_{k}, \\
  I_{yz} = I_{zy} \ &\stackrel{\mathrm{def}}{=}\  -\sum_{k=1}^{N} m_{k} y_{k} z_{k}.
\end{align}</math>

Here <math>I_{xx}</math> denotes the moment of inertia around the <math>x</math>-axis when the objects are rotated around the x-axis, <math>I_{xy}</math> denotes the moment of inertia around the <math>y</math>-axis when the objects are rotated around the <math>x</math>-axis, and so on.

These quantities can be generalized to an object with distributed mass, described by a mass density function, in a similar fashion to the scalar moment of inertia. One then has
<math display="block">\mathbf{I} = \iiint_V \rho(x,y,z) \left( \|\mathbf{r}\|^2 \mathbf{E}_{3} - \mathbf{r}\otimes \mathbf{r}\right)\, dx \, dy \, dz,</math>
where <math>\mathbf{r}\otimes \mathbf{r}</math> is their [[outer product]], '''E'''<sub>3</sub> is the 3×3 [[identity matrix]], and ''V'' is a region of space completely containing the object.

Alternatively it can also be written in terms of the [[Cross product#Conversion to matrix multiplication|angular momentum operator]] <math>[\mathbf r]\mathbf x = \mathbf r\times\mathbf x</math>:
<math display="block">\mathbf{I} = \iiint_V \rho(\mathbf{r}) [\mathbf r]^\textsf{T}[\mathbf r] \, dV = -\iiint_{Q} \rho(\mathbf{r}) [\mathbf r]^2 \, dV </math>

The inertia tensor can be used in the same way as the inertia matrix to compute the scalar moment of inertia about an arbitrary axis in the direction <math>\mathbf{n}</math>,
<math display="block">I_n = \mathbf{n}\cdot\mathbf{I}\cdot\mathbf{n},</math>
where the [[dot product]] is taken with the corresponding elements in the component tensors. A product of inertia term such as <math>I_{12}</math> is obtained by the computation
<math display="block">I_{12} = \mathbf{e}_1\cdot\mathbf{I}\cdot\mathbf{e}_2,</math>
and can be interpreted as the moment of inertia around the <math>x</math>-axis when the object rotates around the <math>y</math>-axis.

The components of tensors of degree two can be assembled into a matrix. For the inertia tensor this matrix is given by,
<math display="block">\begin{align}
\mathbf{I} &= \begin{bmatrix}
    I_{11} & I_{12} & I_{13} \\[1.8ex]
    I_{21} & I_{22} & I_{23} \\[1.8ex]
    I_{31} & I_{32} & I_{33}
  \end{bmatrix} = \begin{bmatrix}
    I_{xx} & I_{xy} & I_{xz} \\[1.8ex]
    I_{yx} & I_{yy} & I_{yz} \\[1.8ex]
    I_{zx} & I_{zy} & I_{zz}
  \end{bmatrix} \\[2ex]
  &= \sum_{k=1}^N \begin{bmatrix}
      m_{k} \left(y_{k}^2 + z_{k}^2\right) &
    - m_{k} x_{k} y_{k} &
    - m_{k} x_{k} z_{k} \\[1ex]
    - m_{k} x_{k} y_{k} &
      m_{k} \left(x_{k}^2 + z_{k}^2\right) &
    - m_{k} y_{k} z_{k} \\[1ex]
    - m_{k} x_{k} z_{k} &
    - m_{k} y_{k} z_{k} &
      m_{k} \left(x_{k}^2 + y_{k}^2\right)
  \end{bmatrix}.
\end{align}
</math>

It is common in rigid body mechanics to use notation that explicitly identifies the <math>x</math>, <math>y</math>, and <math>z</math>-axes, such as <math>I_{xx}</math> and <math>I_{xy}</math>, for the components of the inertia tensor.