(xyz)→(x′y′z′)=Rx(α)⋅(xyz)=(1000cosαsinα0−sinαcosα)⋅(xyz)
If α≪1 we may write
(x′y′z′)≈(10001α0−α1)⋅(xyz)=(1+α⋅tx)⋅(xyz)
where the matrices 1 and tx are defined as
1≡(100010001)tx≡(0000010−10)
In a sense, the matrix tx "generates" a rotation about the x-axis. Likewise, the (generator) matrices for rotations about the y- and z-axis ("y-rotation" and "z-rotation") are
ty≡(001000−100)tz≡(010−100000)
If I understand the Lie-terature correctly, the concept of infinitesimal rotations is one of the key ideas in the study of continuous groups.
Now one might ask how much "deviation from commutativity" is caused by two consecutive infinitesimal rotations? By "deviation from commutativity" I mean the distance between points infinitesimally rotated twice, if the order of the two infinitesimal rotations is exchanged. E.g., for infinitesimal x- and y-rotations we find
(Rx⋅Ry−Ry⋅Rx)⋅(xyz)≈((1+α⋅tx)⋅(1+α⋅ty)−(1+α⋅ty)⋅(1+α⋅tx))⋅(xyz)=α2⋅(tx⋅ty−ty⋅tx)⋅(xyz)which, by actually doing the matrix multiplication, is found to be
α2⋅(tx⋅ty−ty⋅tx)⋅(xyz)=α2⋅tz⋅(xyz)
With the notation
[tx,ty]≡tx⋅ty−ty⋅tx
we may write
[tx,ty]=tz
and likewise
[ty,tz]=tx[tz,tx]=ty
The visual interpretation of equation (1) is illustrated here
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