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 March 1st, 2013, 01:35 PM #1 Newbie   Joined: Mar 2013 Posts: 1 Thanks: 0 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? March 1st, 2013, 08:41 PM #2 Senior Member   Joined: Feb 2013 Posts: 281 Thanks: 0 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 Tags angular, exponential, map, matrices, momentum, rotation Search tags for this page

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