Infinitesimal rotation matrix: Difference between revisions

Revision as of 15:36, 8 September 2024

Revision as of 23:00, 8 March 2025

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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 an infinitesimal three-dimensional rotation about the {{mvar|x}}-axis, a basis element of <math>\mathfrak{so}(3),</math> then

:<math> dL_L_{x} = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{bmatrix} </math>, and <math> I+d\theta 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>{{Harv|Goldstein|Poole|Safko|2002|loc=§4.8}}</ref> It turns out that ''the order in which infinitesimal rotations are applied is irrelevant''.

==Discussion==