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Tuesday, 17 July 2012

Infinitesimal rotations

This animated GIF

was created by repeatedly rotating the vertex points (x,y,z) through a small angle α around the x-axis into the point (x,y,z),
(xyz)(xyz)=Rx(α)(xyz)=(1000cosαsinα0sinαcosα)(xyz)
If α1 we may write
(xyz)(10001α0α1)(xyz)=(1+αtx)(xyz)
where the matrices 1 and tx are defined as
1(100010001)tx(000001010)
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(001000100)tz(010100000)
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
(RxRyRyRx)(xyz)((1+αtx)(1+αty)(1+αty)(1+αtx))(xyz)=α2(txtytytx)(xyz)which, by actually doing the matrix multiplication, is found to be
α2(txtytytx)(xyz)=α2tz(xyz)
With the notation
[tx,ty]txtytytx
we may write
[tx,ty]=tz
and likewise
[ty,tz]=tx[tz,tx]=ty
The visual interpretation of equation (1) is illustrated here

Eight points are rotated about the x-axes, then about the y-axis. The result (blue dots) is compared with the results obtained by the same rotations, however applied in reverse order (black dots). The difference (red line) corresponds to a rotation about the z-axis. Formally, the commutator of the two infinitesimal rotations [tx,ty] is again an infinitesimal rotation, a z-rotation generated by tz.

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