Infinitesimal rotation matrix: Difference between revisions

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An '''infinitesimal rotation matrix''' is a [[skew-symmetric matrix]] where:

#REDIRECT [[Rotation matrix#Infinitesimal rotations]]

* As any [[rotation matrix]] has a single real eigenvalue, which is equal to +1, the corresponding [[eigenvector]] defines the [[rotation axis]].

* Its module defines an infinitesimal [[angular displacement]].

The shape of the matrix is as follows:

{{R to section}}

<math display="block">

A = \begin{pmatrix}

1 & -d\phi_z(t) & d\phi_y(t) \\

d\phi_z(t) & 1 & -d\phi_x(t) \\

-d\phi_y(t) & d\phi_x(t) & 1 \\

\end{pmatrix}

</math>

We can introduce here the associated '''infinitesimal angular displacement tensor''' or '''rotation generator''':

: <math>

d\Phi(t) = \begin{pmatrix}

0 & -d\phi_z(t) & d\phi_y(t) \\

d\phi_z(t) & 0 & -d\phi_x(t) \\

-d\phi_y(t) & d\phi_x(t) & 0 \\

\end{pmatrix}

</math>

Such that its associated rotation matrix is <math>A = I + d\Phi(t)</math>. When it is divided by the time, this will yield the [[angular velocity]] vector.

== Generators of rotations ==

{{Main|Rotation matrix|Rotation group SO(3)|Infinitesimal transformation}}

Suppose we specify an axis of rotation by a unit vector [''x'', ''y'', ''z''], and suppose we have an [[Infinitesimal rotation|infinitely small rotation]] of angle Δ''θ'' about that vector. Expanding the rotation matrix as an infinite addition, and taking the first order approach, the rotation matrix Δ''R'' is represented as:

: <math>\Delta R =

\begin{bmatrix}

1 & 0 & 0 \\

0 & 1 & 0 \\

0 & 0 & 1

\end{bmatrix}

+

\begin{bmatrix}

0 & z & -y \\

-z & 0 & x \\

y & -x & 0

\end{bmatrix}\,\Delta \theta

= I + A\,\Delta\theta.

</math>

A finite rotation through angle ''θ'' about this axis may be seen as a succession of small rotations about the same axis. Approximating Δ''θ'' as ''θ''/''N'', where ''N'' is a large number, a rotation of ''θ'' about the axis may be represented as:

: <math>R = \left(I + \frac{A\theta}{N}\right)^N \approx e^{A\theta}.</math>

It can be seen that Euler's theorem essentially states that ''all'' rotations may be represented in this form. The product ''Aθ'' is the "generator" of the particular rotation, being the vector {{nowrap|(''x'', ''y'', ''z'')}} associated with the matrix ''A''. This shows that the rotation matrix and the [[axis-angle]] format are related by the exponential function.

One can derive a simple expression for the generator ''G''. One starts with an arbitrary plane<ref>in Euclidean space</ref> defined by a pair of perpendicular unit vectors ''a'' and ''b''. In this plane one can choose an arbitrary vector ''x'' with perpendicular ''y''. One then solves for ''y'' in terms of ''x'' and substituting into an expression for a rotation in a plane yields the rotation matrix ''R'', which includes the generator {{nowrap|1=''G'' = ''ba''^T − ''ab''^T}}.

:<math>\begin{align}

x &= a \cos\left( \alpha \right) + b \sin\left( \alpha \right) \\

y &= -a \sin\left( \alpha \right) + b \cos\left( \alpha \right) \\

\cos\left( \alpha \right) &= a^\mathrm{T} x \\

\sin\left( \alpha \right) &= b^\mathrm{T} x \\

y &= -ab^\mathrm{T} x + ba^\mathrm{T} x = \left( ba^\mathrm{T} - ab^\mathrm{T} \right)x \\

\\

x' &= x \cos\left( \beta \right) + y \sin\left( \beta \right) \\

&= \left[ I \cos\left( \beta \right) + \left( ba^\mathrm{T} - ab^\mathrm{T} \right) \sin\left( \beta \right) \right]x \\

\\

R &= I \cos\left( \beta \right) + \left( ba^\mathrm{T} - ab^\mathrm{T} \right) \sin\left( \beta \right) \\

&= I \cos\left( \beta \right) + G \sin\left( \beta \right) \\

\\

G &= ba^\mathrm{T} - ab^\mathrm{T} \\

\end{align}</math>

To include vectors outside the plane in the rotation one needs to modify the above expression for ''R'' by including two [[Projection (linear algebra)|projection operators]] that partition the space. This modified rotation matrix can be rewritten as an [[Matrix exponential#Rotation case|exponential function]].

: <math>\begin{align}

P_{ab} &= -G^2 \\

R &= I - P_{ab} + \left[ I \cos\left( \beta \right) + G \sin\left( \beta \right) \right] P_{ab} = e^{G\beta} \\

\end{align}</math>

Analysis is often easier in terms of these generators, rather than the full rotation matrix. Analysis in terms of the generators is known as the [[Lie algebra]] of the rotation group.

=== Relationship with Lie algebras ===

The matrices in the [[Lie algebra]] are not themselves rotations; the skew-symmetric matrices are derivatives, proportional differences of rotations. An actual "differential rotation", or ''infinitesimal rotation matrix'' has the form

:<math> I + A \, d\theta ~,</math>

where {{math|''dθ''}} is vanishingly small and {{math|''A'' ∈ '''so'''(n)}}, for instance with {{math|1=''A'' = ''L''_''x''}},

: <math> dL_{x} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & -d\theta \\ 0 & d\theta & 1 \end{bmatrix}. </math>

The computation rules are as usual except that infinitesimals of second order are routinely dropped. With these rules, these matrices do not satisfy all the same properties as ordinary finite rotation matrices under the usual treatment of infinitesimals.<ref>{{Harv|Goldstein|Poole|Safko|2002|loc=§4.8}}</ref> It turns out that ''the order in which infinitesimal rotations are applied is irrelevant''. To see this exemplified, consult [[Rotation group SO(3)#Infinitesimal rotations|infinitesimal rotations SO(3)]].

== Exponential map ==

{{main|Rotation group SO(3)#Exponential map|Matrix exponential}}

Connecting the Lie algebra to the Lie group is the [[exponential map (Lie theory)|exponential map]], which is defined using the standard [[matrix exponential]] series for {{math|''e^A''}}<ref>{{Harv|Wedderburn|1934|loc=§8.02}}</ref> For any [[skew-symmetric matrix]] {{mvar|A}}, {{math|exp(''A'')}} is always a rotation matrix.{{efn|Note that this exponential map of skew-symmetric matrices to rotation matrices is quite different from the Cayley transform discussed earlier, differing to 3rd order,

<math>e^{2A} - \frac{I+A}{I-A} = - \frac{2}{3} A^3 +\mathrm{O} (A^4) ~. </math> <br />

Conversely, a [[skew-symmetric matrix]] {{mvar|A}} specifying a rotation matrix through the Cayley map specifies the ''same'' rotation matrix through the map {{math|exp(2 artanh ''A'')}}.}}

An important practical example is the {{math|3 × 3}} case. In [[rotation group SO(3)]], it is shown that one can identify every {{math|''A'' ∈ '''so'''(3)}} with an Euler vector {{math|1='''ω''' = ''θ'' '''u'''}}, where {{math|1='''u''' = (''x'',''y'',''z'')}} is a unit magnitude vector.

By the properties of the identification {{math|'''su'''(2) ≅ '''R'''³}}, {{math|'''u'''}} is in the null space of {{mvar|A}}. Thus, {{math|'''u'''}} is left invariant by {{math|exp(''A'')}} and is hence a rotation axis.

Using [[Rodrigues' rotation formula#Matrix notation|Rodrigues' rotation formula on matrix form]] with {{math|1=''θ'' = {{frac|''θ''|2}} + {{frac|''θ''|2}}}}, together with standard [[List of trigonometric identities#Multiple-angle and half-angle formulae|double angle formulae]] one obtains,

: <math>\begin{align}

\exp( A ) &{}= \exp(\theta(\boldsymbol{u\cdot L}))

= \exp \left( \left[\begin{smallmatrix} 0 & -z \theta & y \theta \\ z \theta & 0&-x \theta \\ -y \theta & x \theta & 0 \end{smallmatrix}\right] \right)= \boldsymbol{I} + 2\cos\frac{\theta}{2}\sin\frac{\theta}{2}~\boldsymbol{u\cdot L} + 2\sin^2\frac{\theta}{2} ~(\boldsymbol{u\cdot L} )^2 ,

\end{align}</math>

This is the matrix for a rotation around axis {{math|'''u'''}} by the angle {{mvar|θ}} in half-angle form. For full detail, see [[Rotation group SO(3)#Exponential map|exponential map SO(3)]].

Notice that for infinitesimal angles second order terms can be ignored and remains {{math|1=exp(''A'') = ''I'' + ''A''}}

==See also==

*[[Euler's rotation theorem#Generators of rotations{{!}}Generators of rotations]]

*[[Rotation matrix#Infinitesimal rotations{{!}}Infinitesimal rotations]]

*[[Infinitesimal strain theory#Infinitesimal rotation tensor{{!}}Infinitesimal rotation tensor]]

*[[Rotation group SO(3)#Infinitesimal rotations]]

==Notes==

{{notelist}}

==References==

{{reflist}}

[[Category:Rotation]]

[[Category:Infinitesimals]]