Infinitesimal rotation matrix: Difference between revisions

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An '''infinitesimal rotation matrix''' or '''differential rotation matrix''' hasis thea form:[[matrix (mathematics)|matrix]] representing an infinitely slight [[rotation]].

:<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>

While a [[rotation matrix]] is an [[orthogonal matrix]] <math>R^\mathsf{T} = R^{-1}</math> representing an element of <math>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> which is not itself a rotation matrix.

While the [[matrices]] in the [[Lie algebra]] are not themselves rotations, the [[skew-symmetric matrices]] are [[derivative (mathematics)|derivatives]], proportional differences of rotations.

An infinitesimal rotation matrix has the form

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

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

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

:<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==