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March 1st, 2013, 01:35 PM   #1
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Rotation matrices and exponential map for angular momentum

Hi guys,

I'm having some trouble with rotation matrices. Basically, this is the problem:

Let R(na) be a matrix in SO(3), where na specifies the rotation the matrix represents. Namlely, n is the unit vector about which the rotation is performed, and a is the angle of the rotation. Let {J1,J2,J3} be the standard basis of the Lie algebra so(3) (which is the set of 3x3 skew-symmetric matrices). Then prove

R(na) = exp[a * ? n_i J_i].


I know that exp[a_1 J_1] is the rotation of a_1 degrees about the x-axis, and similarly for exp[a_2 J_2] and exp[a_3 J_3].
I also know that the exponent map from so(3) to SO(3) is surjective. This gives that any R(na) can be written as

R(na)= exp[X(na)]

for some X(na) in so(3). I know I'm also supposed to use the fact that the Ji form a basis for so(3), but I can't exactly prove the problem above. Can anyone help?
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March 1st, 2013, 08:41 PM   #2
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Re: Rotation matrices and exponential map for angular moment

The infinitesimal generator of a Lie group is defined by

Otherweise saying we consider the element infinitesimally near to the identity element, i.e.


The rotation matrix about the z axis is well-known, derivating the you can show


You can manually show that it's true on the rest axises


Now consider the general rotation, i.e. an infinitesimal rotation about n axis. Heuristically we can say

or


Comparing this with the second definition of infinitesimal generator shows that

or reexpressing with the axis generators


Thus the general rotation is
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