“Infinitesimal Rotation”的概念、定义、习题解题方法及翻译-数学辞典

单词

Infinitesimal Rotation

释义

Infinitesimal Rotation

An infinitesimal transformation of a Vector is given by

(1)

where the Matrix

is infinitesimal and I is the Identity Matrix. (Note that the infinitesimaltransformation may not correspond to an inversion, since inversion is a discontinuous process.) TheCommutativity of infinitesimal transformations

and

is established by theequivalence of

(2)

(3)

Now let

(4)

The inverse

is then

, since

(5)

Since we are defining our infinitesimal transformation to be a rotation, Orthogonalityof Rotation Matricesrequires that

(6)

but

(7)

(8)

and the infinitesimal rotation is Antisymmetric. It must therefore have a Matrix of the form

(9)

The differential change in a vector

upon application of the Rotation Matrix is then

(10)

Writing in Matrix form,


			(11)

			(12)

Therefore,

(13)

where

(14)

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

(15)

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

(16)

See also Acceleration, Euler Angles, Rotation, Rotation Matrix, Rotation Operator

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