Timeline for Prove that infinitesmal angular displacements commute & non-infinitesimal ones don't?
Current License: CC BY-SA 3.0
4 events
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| Aug 30, 2016 at 10:08 | comment | added | Xasel | Respected Sir,thank you for taking out your precious time to help me out .Can you please me a little simpler explanation as I haven't studied Linear ALgebra(that's why i specified my background u someone edited that out).Can I prove that same using only Calc + Algebra(whatever taught at highchool level). | |
| Aug 29, 2016 at 18:22 | comment | added | Gary Godfrey | @Sean - Yes, you are absolutely correct. I interpreted the question as asking specifically for the commutator of angular displacements, so I kept it simple and didn't introduce generators. As you suggest, the group of rotations and the Lie algebra of its generators would be very useful/enlightening for the questioner to learn. | |
| Aug 29, 2016 at 16:31 | comment | added | Sean E. Lake | Small correction - the commutator is relevant to the generators, not the group elements (the rotation matrices). If I add two rotation matrices, I don't get a rotation matrix back. The quantity you want to look at for the test of whether a group is abelian is: $$B^{-1}A^{-1}BA,$$ ie try to walk in a closed square. In other words, if you rewrite the matrices $$M_i = I + \theta_i J_i$$ then $$M_x^{-1}M_y^{-1}M_xM_y \approx I + \theta_x\theta_y [J_x, J_y]$$ to first order in both of the $\theta$s with $I$ the identity matrix. | |
| Aug 28, 2016 at 22:05 | history | answered | Gary Godfrey | CC BY-SA 3.0 |