Editing Moment of inertia (section)

== Inertia matrix in different reference frames ==
The use of the inertia matrix in Newton's second law assumes its components are computed relative to axes parallel to the inertial frame and not relative to a body-fixed reference frame.<ref name="Kane"/><ref name="Goldstein"/> This means that as the body moves the components of the inertia matrix change with time. In contrast, the components of the inertia matrix measured in a body-fixed frame are constant.

=== Body frame ===
Let the body frame inertia matrix relative to the center of mass be denoted <math>\mathbf{I}_\mathbf{C}^B</math>, and define the orientation of the body frame relative to the inertial frame by the rotation matrix <math>\mathbf{A}</math>, such that,
<math display="block">\mathbf{x} = \mathbf{A}\mathbf{y},</math>
where vectors <math>\mathbf{y}</math> in the body fixed coordinate frame have coordinates <math>\mathbf{x}</math> in the inertial frame. Then, the inertia matrix of the body measured in the inertial frame is given by
<math display="block">\mathbf{I}_\mathbf{C} = \mathbf{A} \mathbf{I}_\mathbf{C}^B \mathbf{A}^\mathsf{T}.</math>

Notice that <math>\mathbf{A}</math> changes as the body moves, while <math>\mathbf{I}_\mathbf{C}^B</math> remains constant.

=== Principal axes ===
Measured in the body frame, the inertia matrix is a constant real symmetric matrix. A real symmetric matrix has the [[eigendecomposition of a matrix|eigendecomposition]] into the product of a rotation matrix <math>\mathbf{Q}</math> and a diagonal matrix <math>\boldsymbol{\Lambda}</math>, given by
<math display="block">\mathbf{I}_\mathbf{C}^B = \mathbf{Q}\boldsymbol{\Lambda}\mathbf{Q}^\mathsf{T},</math>
where
<math display="block">\boldsymbol{\Lambda} = \begin{bmatrix}
  I_1 & 0 & 0 \\
  0 & I_2 & 0 \\
  0 & 0 & I_3
\end{bmatrix}.</math>

The columns of the rotation matrix <math>\mathbf{Q}</math> define the directions of the principal axes of the body, and the constants <math>I_1</math>, <math>I_2</math>, and <math>I_3</math> are called the '''principal moments of inertia'''. This result was first shown by [[James Joseph Sylvester|J. J. Sylvester (1852)]], and is a form of [[Sylvester's law of inertia]].<ref name=syl852>{{cite journal |author=Sylvester, J J | title=A demonstration of the theorem that every homogeneous quadratic polynomial is reducible by real orthogonal substitutions to the form of a sum of positive and negative squares | journal=Philosophical Magazine |series=4th Series| volume=4 | issue=23 | pages=138–142 | year=1852 |  url=http://www.maths.ed.ac.uk/~aar/sylv/inertia.pdf | doi= 10.1080/14786445208647087 | access-date=June 27, 2008}}</ref><ref name=norm>{{cite book| author=Norman, C.W.| title=Undergraduate algebra | publisher=[[Oxford University Press]] | pages=360–361 | year=1986 |  isbn=0-19-853248-2 }}</ref> When the body has an axis of symmetry (sometimes called the '''figure axis''' or '''axis of figure''') then the other two moments of inertia will be identical and any axis perpendicular to the axis of symmetry will be a principal axis.

A toy [[Spinning top|top]] is an example of a rotating rigid body, and the word ''top'' is used in the names of types of rigid bodies.  When all principal moments of inertia are distinct, the principal axes through [[center of mass]] are uniquely specified and the rigid body is called an '''asymmetric top'''. If two principal moments are the same, the rigid body is called a '''symmetric top''' and there is no unique choice for the two corresponding principal axes. If all three principal moments are the same, the rigid body is called a '''spherical top''' (although it need not be spherical) and any axis can be considered a principal axis, meaning that the moment of inertia is the same about any axis.

The principal axes are often aligned with the object's symmetry axes. If a rigid body has an axis of symmetry of order <math>m</math>, meaning it is symmetrical under rotations of {{math|[[turn (geometry)|360°]]/''m''}} about the given axis, that axis is a principal axis. When <math>m > 2</math>, the rigid body is a symmetric top. If a rigid body has at least two symmetry axes that are not parallel or perpendicular to each other, it is a spherical top, for example, a cube or any other [[Platonic solid]].

The [[motion (physics)|motion]] of [[vehicle]]s is often described in terms of [[yaw, pitch, and roll]] which usually correspond approximately to rotations about the three principal axes. If the vehicle has bilateral symmetry then one of the principal axes will correspond exactly to the transverse (pitch) axis.

A practical example of this mathematical phenomenon is the routine automotive task of [[Tire balance|balancing a tire]], which basically means adjusting the distribution of mass of a car wheel such that its principal axis of inertia is aligned with the axle so the wheel does not wobble.

[[Rotational spectroscopy#Classification of molecular rotors|Rotating molecules are also classified]] as asymmetric, symmetric, or spherical tops, and the structure of their [[Rotational spectroscopy|rotational spectra]] is different for each type.

=== Ellipsoid ===
[[File:Triaxial Ellipsoid.jpg|thumb|right|An ellipsoid with the semi-principal diameters labelled <math>a</math>, <math>b</math>, and <math>c</math>.]]
The moment of inertia matrix in body-frame coordinates is a quadratic form that defines a surface in the body called [[Poinsot's ellipsoid]].<ref>
{{cite book
 |first=Matthew T. |last=Mason
 |url=https://books.google.com/books?id=Ngdeu3go014C
 |title=Mechanics of Robotics Manipulation
 |publisher=MIT Press
 |year=2001
 |isbn=978-0-262-13396-8
 |access-date=November 21, 2014
}}</ref> Let <math>\boldsymbol{\Lambda}</math> be the inertia matrix relative to the center of mass aligned with the principal axes, then the surface
<math display="block">\mathbf{x}^\mathsf{T}\boldsymbol{\Lambda}\mathbf{x} = 1,</math>
or
<math display="block">I_1x^2 + I_2y^2 + I_3z^2 =1,</math>
defines an [[ellipsoid]] in the body frame. Write this equation in the form,
<math display="block"> \left(\frac{x}{1/\sqrt{I_1}}\right)^2 + \left(\frac{y}{1/\sqrt{I_2}}\right)^2 + \left(\frac{z}{1/\sqrt{I_3}}\right)^2 = 1,</math>
to see that the semi-principal diameters of this ellipsoid are given by
<math display="block">a = \frac{1}{\sqrt{I_1}}, \quad b=\frac{1}{\sqrt{I_2}}, \quad c=\frac{1}{\sqrt{I_3}}.</math>

Let a point <math>\mathbf{x}</math> on this ellipsoid be defined in terms of its magnitude and direction, <math>\mathbf{x} = \|\mathbf{x}\|\mathbf{n}</math>, where <math>\mathbf{n}</math> is a unit vector. Then the relationship presented above, between the inertia matrix and the scalar moment of inertia <math>I_\mathbf{n}</math> around an axis in the direction <math>\mathbf{n}</math>, yields
<math display="block">\mathbf{x}^\mathsf{T}\boldsymbol{\Lambda}\mathbf{x} = \|\mathbf{x}\|^2\mathbf{n}^\mathsf{T}\boldsymbol{\Lambda}\mathbf{n} = \|\mathbf{x}\|^2 I_\mathbf{n} = 1. </math>

Thus, the magnitude of a point <math>\mathbf{x}</math> in the direction <math>\mathbf{n}</math> on the inertia ellipsoid is
<math display="block"> \|\mathbf{x}\| = \frac{1}{\sqrt{I_\mathbf{n}}}.</math>
<!---duplicated above
=== Parallel axis theorem for the inertia matrix ===
{{Main|Parallel axis theorem}}
It is useful to note here that if the moment of inertia matrix or tensor is relative to the [[center of mass]], then it can be determined relative to any other reference point in the body using the parallel axis theorem.  If [I{{sub|C}}{{sup|B}}] is the moment of inertia matrix in the body frame relative to the center of mass '''C''', then the moment of inertia matrix [I{{sub|R}}{{sup|B}}] in the same frame but relative to a different point '''R''' is given by
<math display="block">[I_R^B] = [I_C^B] - M[d]^2, </math>
where M is the mass of the body, and [d] is the skew-symmetric matrix obtained from the vector '''d''' = '''R'''&nbsp;−&nbsp;'''C'''.

The tensor form of the parallel axis theorem is given by
<math display="block"> \mathbf{I}_R^B = \mathbf{I}_C^B + M((\mathbf{d} \cdot \mathbf{d}) \mathbf{E} - \mathbf{d} \otimes \mathbf{d}).
</math>
-->