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Infinitesimal Rotation


An infinitesimal transformation of a vector r is given by

 r^'=(I+e)r,
(1)

where the matrix e is infinitesimal and I is the identity matrix. (Note that the infinitesimal transformation may not correspond to an inversion, since inversion is a discontinuous process.) The commutativity of infinitesimal transformations e_1 and e_2 is established by the equivalence of

(I+e_1)(I+e_2)=I^2+e_1I+Ie_2+e_1e_2
(2)
 approx I+e_1+e_2
(3)
(I+e_2)(I+e_1)=I^2+e_2I+Ie_1+e_2e_1
(4)
 approx I+e_2+e_1.
(5)

Now let

 A=I+e.
(6)

The inverse A^(-1) is then I-e, since

AA^(-1)=(I+e)(I-e)
(7)
=I^2-e^2
(8)
 approx I.
(9)

Since we are defining our infinitesimal transformation to be a rotation, orthogonality of rotation matrices requires that

 A^(T)=A^(-1),
(10)

but

A^(-1)=I-e
(11)
(I+e)^(T)=I^(T)+e^(T)
(12)
=I+e^(T),
(13)

so e=-e^(T) and the infinitesimal rotation is antisymmetric. It must therefore have a matrix of the form

 e=[0 dOmega_3 -dOmega_2; -dOmega_3 0 dOmega_1; dOmega_2 -dOmega_1 0].
(14)

The differential change in a vector r upon application of the rotation matrix is then

 dr=r^'-r=(I+e)r-r=er.
(15)

Writing in matrix form,

dr=[0 dOmega_3 -dOmega_2; -dOmega_3 0 dOmega_1; dOmega_2 -dOmega_1 0][x; y; z]
(16)
=[ydOmega_3-zdOmega_2; zdOmega_1-xdOmega_3; xdOmega_2-ydOmega_1]
(17)
=(ydOmega_3-zdOmega_2)x^^+(zdOmega_1-xdOmega_3)y^^+(xdOmega_2-ydOmega_1)z^^
(18)
=rxdOmega.
(19)

Therefore,

 ((dr)/(dt))_(rotation, body)=rx(dOmega)/(dt)=rxomega,
(20)

where

 omega=(dOmega)/(dt)=n^^(dphi)/(dt).
(21)

The total rotation observed in the stationary frame will be a sum of the rotational velocity and the velocity in the rotating frame. However, note that an observer in the stationary frame will see a velocity opposite in direction to that of the observer in the frame of the rotating body, so

 ((dr)/(dt))_(space)=((dr)/(dt))_(body)+omegaxr.
(22)

This can be written as an operator equation, known as the rotation operator, defined as

 (d/(dt))_(space)=(d/(dt))_(body)+omegax.
(23)

See also

Acceleration, Euler Angles, Infinitesimal Matrix Change, Rotation, Rotation Matrix, Rotation Operator

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Cite this as:

Weisstein, Eric W. "Infinitesimal Rotation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/InfinitesimalRotation.html

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