Infinitesimal Rotation

An infinitesimal transformation of a vector r is given by


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

 approx I+e_1+e_2
 approx I+e_2+e_1.

Now let


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

 approx I.

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




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].

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


Writing in matrix form,

dr=[0 dOmega_3 -dOmega_2; -dOmega_3 0 dOmega_1; dOmega_2 -dOmega_1 0][x; y; z]
=[ydOmega_3-zdOmega_2; zdOmega_1-xdOmega_3; xdOmega_2-ydOmega_1]


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



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


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


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.

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