Infinitesimal rotation matrix: Difference between revisions

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An '''infinitesimal rotation matrix''' or '''differential rotation matrix''' is a [[matrix (mathematics)|matrix]] representing an [[infinitesimal|infinitely]] small [[rotation]].

While a [[rotation matrix]] is an [[orthogonal matrix]] <math>R^\mathsf{T} = R^{-1}</math> representing an element of <math>\mathrm{SO}(n)</math> (the [[special orthogonal group]]), the [[differential (mathematics)|differential]] of a rotation is a [[skew-symmetric matrix]] <math>A^\mathsf{T} = -A</math> in the [[tangent space]] <math>\mathfrak{so}(n)</math> (the [[special orthogonal Lie algebra]]), which is not itself a rotation matrix.

An infinitesimal rotation matrix has the form

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

where <math>I</math> is the identity matrix, <math>d\theta</math> is vanishingly small, and <math>{{tmath| A \in \mathfrak{so}(n) }}.</math>

For example, if <math>{{tmath|1= A = L_x }},</math> representing an infinitesimal three-dimensional rotation about the {{mvar|x}}-axis, a basis element of <math>{{tmath| \mathfrak{so}(3) }},</math> then

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

: <math> dL_L_{x} = \begin{bmatrix} 10 & 0 & 0 \\ 0 & 10 & -d\theta1 \\ 0 & d\theta1 & 10 \end{bmatrix}. ,</math>

and

: <math> dA_{I+d\mathbftheta L_{x}} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & -d\theta \\ 0 & d\theta & 1 \end{bmatrix} .</math>

The computation rules for infinitesimal rotation matrices are asthe usual ones 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>{{Harvharv|Goldstein|Poole|Safko|2002|loc=§4.8}}</ref> It turns out that ''the order in which infinitesimal rotations are applied is irrelevant''.

where <math>I</math> is the identity matrix, <math>d\theta</math> is vanishingly small, and <math>A \in \mathfrak{so}(n).</math>

== Discussion ==

For example, if <math>A = L_x,</math> representing an infinitesimal three-dimensional rotation about the {{mvar|x}}-axis, a basis element of <math>\mathfrak{so}(3),</math>

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

The computation rules for infinitesimal rotation matrices 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''.

==Discussion==

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

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

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</math>

=== Associated quantities ===

WeAssociated canto introducean hereinfinitesimal therotation associatedmatrix <math>A</math> is an ''[[infinitesimal rotation tensor]]'' <math>d\Phi(t) = A - I</math>:

: <math display="block">

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

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

Line 38 ⟶ 37:

\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.

Dividing it by the time difference yields the ''[[angular velocity tensor]]'':

==Order of rotations==

: <math>

\Omega = \frac{d\Phi(t)}{dt} = \begin{pmatrix}

0 & -\omega_z(t) & \omega_y(t) \\

\omega_z(t) & 0 & -\omega_x(t) \\

-\omega_y(t) & \omega_x(t) & 0 \\

\end{pmatrix}

</math>

== Order of rotations ==

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> To understand what this means, consider

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> To understand what this means, consider

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

: <math> dA_{\mathbf{x}}^2 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 - d\theta^2 & -2dd\theta \\ 0 & 2\,d\theta & 1 - d\theta^2 \end{bmatrix}.</math>

First, test the orthogonality condition, {{math|1=''Q''^T''Q'' = ''I''}}. The product is

: <math> dA_\mathbf{x}^\textsf{T} \, dA_\mathbf{x} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 + d\theta^2 & 0 \\ 0 & 0 & 1 + d\theta^2 \end{bmatrix},</math>

differing from an identity matrix by second -order infinitesimals, discarded here. So, to first order, an infinitesimal rotation matrix is an orthogonal matrix.

:<math> dA_\mathbf{x}^\textsf{T} \, dA_\mathbf{x} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 + d\theta^2 & 0 \\ 0 & 0 & 1 + d\theta^2 \end{bmatrix},</math>

differing from an identity matrix by second order infinitesimals, discarded here. So, to first order, an infinitesimal rotation matrix is an orthogonal matrix.

Next, examine the square of the matrix,

: <math> dA_{\mathbf{yx}}^2 = \begin{bmatrix} 1 & 0 & d\phi0 \\ 0 & 1 - d\theta^2 & 0-2d\theta \\ -d\phi0 & 02\,d\theta & 1 - d\theta^2 \end{bmatrix}.</math>

Again discarding second -order effects, note that the angle simply doubles. This hints at the most essential difference in behavior, which we can exhibit with the assistance of a second infinitesimal rotation,

:<math> dA_{\mathbf{x}}^2 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 - d\theta^2 & -2d\theta \\ 0 & 2\,d\theta & 1 - d\theta^2 \end{bmatrix}.</math>

: <math>dA_\mathbf{y} = \begin{bmatrix} 1 & 0 & d\phi \\ 0 & 1 & 0 \\ -d\phi & 0 & 1 \end{bmatrix}.</math>

Compare the products {{math|''dA''_'''x'''&nbsp;''dA''_'''y'''}} to {{math|''dA''_'''y'''&nbsp;''dA''_'''x'''}},

Again discarding second order effects, note that the angle simply doubles. This hints at the most essential difference in behavior, which we can exhibit with the assistance of a second infinitesimal rotation,

: <math>\begin{align}

:<math>dA_\mathbf{y} = \begin{bmatrix} 1 & 0 & d\phi \\ 0 & 1 & 0 \\ -d\phi & 0 & 1 \end{bmatrix}.</math>

Compare the products {{math|''dA''_'''x'''&nbsp;''dA''_'''y'''}} to {{math|''dA''_'''y'''''dA''_'''x'''}},

:<math>\begin{align}

dA_{\mathbf{x}}\,dA_{\mathbf{y}} &= \begin{bmatrix} 1 & 0 & d\phi \\ d\theta\,d\phi & 1 & -d\theta \\ -d\phi & d\theta & 1 \end{bmatrix} \\

dA_{\mathbf{y}}\,dA_{\mathbf{x}} &= \begin{bmatrix} 1 & d\theta\,d\phi & d\phi \\ 0 & 1 & -d\theta \\ -d\phi & d\theta & 1 \end{bmatrix}. \\

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Since <math>d\theta \, d\phi</math> is second-order, we discard it: thus, to first order, multiplication of infinitesimal rotation matrices is ''commutative''. In fact,

: <math> dA_{\mathbf{x}}\,dA_{\mathbf{y}} = dA_{\mathbf{y}}\,dA_{\mathbf{x}},</math>

:<math> dA_{\mathbf{x}}\,dA_{\mathbf{y}} = dA_{\mathbf{y}}\,dA_{\mathbf{x}},</math>

again to first order. In other words, {{em|the order in which infinitesimal rotations are applied is irrelevant}}.

This useful fact makes, for example, derivation of rigid body rotation relatively simple. But one must always be careful to distinguish (the first -order treatment of) these infinitesimal rotation matrices from both finite rotation matrices and from Lie algebra elements. When contrasting the behavior of finite rotation matrices in the BCH[[Baker–Campbell–Hausdorff formula]] above with that of infinitesimal rotation matrices, where all the commutator terms will be second -order infinitesimals, one finds a bona fide vector space. Technically, this dismissal of any second -order terms amounts to [[Groupgroup contraction]].

== 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}

Line 95:

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 = \lim_{N\to\infty} \left(I + \frac{A\theta}{N}\right)^N \approx= e^{\exp(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}

:<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^\mathrmmathsf{T} x \\

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

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

\\

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

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

\\

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

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

\\

G &= ba^\mathrmmathsf{T} - ab^\mathrmmathsf{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^{\exp(G\beta}) \\

\end{align}</math>

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== 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|exp(''e^A'')}}<ref>{{Harvharv|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''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'' = {{fracsfrac|''θ''|2}} + {{fracsfrac|''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 \boldsymbol{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 \boldsymbol{L} + 2\sin^2\frac{\theta}{2} ~(\boldsymbol{u} \cdot \boldsymbol{L} )^2 ,

\end{align}</math>

This is the matrix for a rotation around axis {{math|'''u'''}} by the angle {{mvarmath|''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''}}

== Relationship to skew-symmetric matrices ==

Skew-symmetric matrices over the field of real numbers form the [[tangent space]] to the real [[orthogonal group]] <math>\mathrm{O}(n)</math> at the identity matrix; formally, the [[special orthogonal Lie algebra]]. In this sense, then, skew-symmetric matrices can be thought of as ''infinitesimal rotations''.

Another way of saying this is that the space of skew-symmetric matrices forms the [[Lie algebra]] <math>\mathfrak{o}(n)</math> of the [[Lie group]] <math>{{tmath| \mathrm{O}(n) }}.</math> The Lie bracket on this space is given by the [[commutator]]:

: <math>[A, B] = AB - BA.\,</math>

:<math>[A, B] = AB - BA.\,</math>

It is easy to check that the commutator of two skew-symmetric matrices is again skew-symmetric:

: <math>\begin{align}

{[}A, B{]}^\textsf{T} &= B^\textsf{T} A^\textsf{T} - A^\textsf{T} B^\textsf{T} \\

Line 161 ⟶ 157:

The [[matrix exponential]] of a skew-symmetric matrix <math>A</math> is then an [[orthogonal matrix]] <math>R</math>:

: <math>R = \exp(A) = \sum_{n=0}^\infty \frac{A^n}{n!}.</math>

The image of the [[exponential map (Lie theory)|exponential map]] of a Lie algebra always lies in the [[Connected space|connected component]] of the Lie group that contains the identity element. In the case of the Lie group {{tmath| \mathrm{O}(n) }}, this connected component is the [[special orthogonal group]] {{tmath| \mathrm{SO}(n) }}, consisting of all orthogonal matrices with determinant 1. So <math>R = \exp(A)</math> will have determinant&nbsp;+1. Moreover, since the exponential map of a connected compact Lie group is always surjective, it turns out that ''every'' orthogonal matrix with unit determinant can be written as the exponential of some skew-symmetric matrix.

:<math>R = \exp(A) = \sum_{n=0}^\infty \frac{A^n}{n!}.</math>

The image of the [[exponential map (Lie theory)|exponential map]] of a Lie algebra always lies in the [[Connected space|connected component]] of the Lie group that contains the identity element. In the case of the Lie group <math>O(n),</math> this connected component is the [[special orthogonal group]] <math>SO(n),</math> consisting of all orthogonal matrices with determinant 1. So <math>R = \exp(A)</math> will have determinant&nbsp;+1. Moreover, since the exponential map of a connected compact Lie group is always surjective, it turns out that ''every'' orthogonal matrix with unit determinant can be written as the exponential of some skew-symmetric matrix. In the particular important case of dimension <math>n=2,</math> the exponential representation for an orthogonal matrix reduces to the well-known [[complex number#Polar form|polar form]] of a complex number of unit modulus. Indeed, if <math>{{tmath|1= n=2 }},</math> a special orthogonal matrix has the form

: <math>\begin{bmatrix}

a & -b \\

b & \,a

\end{bmatrix},</math>

with <math>{{tmath|1= a^2 + b^2 = 1</math> }}. Therefore, putting <math>a = \cos\theta</math> and <math>{{tmath|1= b = \sin\theta }},</math> it can be written

: <math>\begin{bmatrix}

with <math>a^2 + b^2 = 1</math>. Therefore, putting <math>a = \cos\theta</math> and <math>b = \sin\theta,</math> it can be written

:<math>\begin{bmatrix}

\cos\,\theta & -\sin\,\theta \\

\sin\,\theta & \,\cos\,\theta

Line 179 ⟶ 175:

\end{bmatrix}\right),

</math>

which corresponds exactly to the polar form <math>\cos \theta + i \sin \theta = e^{\exp(i \theta})</math> of a complex number of unit modulus.

In 3 dimensions, the matrix exponential is [[Rodrigues' rotation formula#Matrix notation|Rodrigues' rotation formula in matrix notation]], and when expressed via the [[Euler-Rodrigues formula]], the algebra of its four parameters [[Euler-Rodrigues formula#Connection with quaternions|gives rise to quaternions]].

which corresponds exactly to the polar form <math>\cos \theta + i \sin \theta = e^{i \theta}</math> of a complex number of unit modulus.

The exponential representation of an orthogonal matrix of order <math>n</math> can also be obtained starting from the fact that in dimension <math>n</math> any special orthogonal matrix <math>R</math> can be written as <math>{{tmath|1= R = QSQ^\textsf{T} }},</math> where <math>Q</math> is orthogonal and S is a [[block matrix#Block diagonal matrix|block diagonal matrix]] with <math display="inline">\lfloor n/2\rfloor</math> blocks of order&nbsp;2, plus one of order 1 if <math>n</math> is odd; since each single block of order 2 is also an orthogonal matrix, it admits an exponential form. Correspondingly, the matrix&nbsp;''S'' writes as exponential of a skew-symmetric block matrix <math>\Sigma</math> of the form above, <math>{{tmath|1= S = \exp(\Sigma) }},</math> so that <math>{{tmath|1= R = Q\exp(\Sigma)Q^\textsf{T} = \exp(Q\Sigma Q^\textsf{T}) }},</math> exponential of the skew-symmetric matrix <math>{{tmath| Q\Sigma Q^\textsf{T} }}.</math> Conversely, the surjectivity of the exponential map, together with the above-mentioned block-diagonalization for skew-symmetric matrices, implies the block-diagonalization for orthogonal matrices.

== 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]]

* [[Infinitesimal transformation]]

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

== Notes ==

{{notelist}}

== References ==

{{reflist}}

== Sources ==

* {{Citationcitation |last1=Goldstein |first1=Herbert |author1-link=Herbert Goldstein |author2-link=Charles P. Poole |last2=Poole |first2=Charles P. |last3=Safko |first3=John L. |year=2002<!-- January 15 --> |title=Classical Mechanics |edition=third |publisher=[[Addison Wesley]] |isbn=978-0-201-65702-9}}

* {{Citationcitation

| last=Wedderburn | first=Joseph H. M. | author-link=Joseph Wedderburn

| first=Joseph H. M.

| author-link=Joseph Wedderburn

| year=1934

| title=Lectures on Matrices

Line 209 ⟶ 204:

| url=https://scholar.google.co.uk/scholar?hl=en&lr=&q=author%3AWedderburn+intitle%3ALectures+on+Matrices&as_publication=&as_ylo=1934&as_yhi=1934&btnG=Search

}}

[[Category:Rotation]]