p09fl7 - Lecture 7 Rotations II(17 Sep 08 A Complete L6.B 1...

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Unformatted text preview: Lecture 7: Rotations II (17 Sep 08) A. Complete L6.B 1. Express the differential of the dot products as: d ˆ e i · ˆ e i = 0 = 2ˆ e i · d ˆ e i d ˆ e i · ˆ e j = 0 = ˆ e i · d ˆ e j + ˆ e j · d ˆ e i ⇒ ˆ e i · d ˆ e j =- ˆ e j · d ˆ e i 2. Expand the differentials in projections onto orthogonal basis vectors: d ˆ e i = X j d Ω ij ˆ e j The dot product with ˆ e i implies d Ω ii = 0 and the i,j combinations give d Ω ij =- d Ω ji . Define d Ω 12 = d Ω 3 ; d Ω 23 = d Ω 1 ; d Ω 31 = d Ω 2 . Hence: d ˆ e 1 = d Ω 12 ˆ e 2 + d Ω 13 ˆ e 3 = d Ω 3 ˆ e 2- d Ω 2 ˆ e 3 d ˆ e 2 = d Ω 21 ˆ e 1 + d Ω 23 ˆ e 3 =- d Ω 1 ˆ e 2 + d Ω 1 ˆ e 3 d ˆ e 3 = d Ω 31 ˆ e 1 + d Ω 32 ˆ e 2 = d Ω 2 ˆ e 1- d Ω 1 ˆ e 2 3. Define d ~ Ω = d Ω 1 ˆ e 1 + d Ω 2 ˆ e 2 + d Ω 3 ˆ e 3 and ˆ e 1 × ˆ e 2 = ˆ e 3 (and cyclic) d ~ Ω × ˆ e 1 = [ d Ω 2 ˆ e 2 + d Ω 3 ˆ e 3 ] × ˆ e 1 = d Ω 3 ˆ e 2- d Ω 2 ˆ e 3 4. Then the signs agree with differential of 6.A: d r = φ ˆ n × r = d ~ Ω × r ....
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This note was uploaded on 09/29/2009 for the course PHYS 711 taught by Professor Bruch during the Fall '09 term at University of Wisconsin.

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p09fl7 - Lecture 7 Rotations II(17 Sep 08 A Complete L6.B 1...

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