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notes_rotations_ver2011

# notes_rotations_ver2011 - 5 5.1 Rotations Rotation matrix...

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5 Rotations 5.1 Rotation matrix The rotation operator ˆ R ( t ) has a matrix R ij ( t ) defined so that if a vector a = a i e i (where e i are the basis vectors in the lab frame) is rotated into vector b = b i e i , then b = ˆ R a , b i = R ij a j The rotated basis vectors e i ( t ) = ˆ R e i satisfy e i ( t ) = e j R ji ( t ) (2) e i = e j ( t ) ( R 1 ( t ) ) ji (3) To memorize the above, recall that when a b by the same rotation which takes e i e i b = e i b i = e j a j e i b i = e i R ij a j = e j a j The requirement of length conservation defines the rotation. It can be expressed as ( ⃗e i ( t ) · e j ( t )) = δ ij . Of course, the lab-frame vectors also satisfy ( e i 0 · e j 0 ) = δ ij . From (2) we have δ ij = ( e i ( t ) · e j ( t )) = R mi R nj ( e m · e n ) = R mi R nj δ mn = R mi R mj In matrix notation that means R T R = E, or alternatively R 1 = R T . 5.2 Instantaneous angular velocity Let us consider any vector r ( t ) = c i e i ( t ) = c i ( t ) e i that is fixed in the rotating body ( c i are time- independent). At t = 0 the body has not yet rotated, so e i (0) = e i and c i (0) = c i . Rotation R ( t ) transforms vector r (0) = c i e i into r ( t ) = c i e i , so c i ( t ) = R ij ( t ) c j Differentiating w.r.t. time we get ˙ c i = ˙ R ij c j = ˙ R ij R ( 1) jk c k = X ik c k , where we defined X jk = ˙ R ji ( R 1 ) ik = ˙ R ji ( R T ) ik , so X = ˙ RR T . First, we prove that X is an antisymmetric matrix. From RR T = E one gets ˙ RR T + R ˙ R T = 0 , ˙ RR T = ( ˙ RR T ) T , X = X T . Since X is antisymmetric, it has a form X = 0 x 12 x 13 x 12 0 x 23 x 13 x 23 0 with only three independent elements. 7

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The velocity of the point described by r can be determined as ˙ r = ˙ e i c i = e j ˙ R ji c i = e j ˙ R ji ( R 1 ) ik c k = e j X jk c k Note that we made a substitution ˙ e i = e j ˙ R ji . This is only true for constant e j . Later we will study the subject of rotation in rotating reference frame, where e j are time-dependent and expressions become more complicated. Now we want to prove that v can be given by the formula v = [ ⃗ω × r ] (4) with some vector ⃗ω . This equation can be translated into a matrix form. Expand ⃗ω = ω i e i ( t ). Then v = [ ω i e i × c j e j ] = ω i c j [ e i × e j ] = ω i c j ϵ ijk e k = ( ϵ ijk ω j c k ) e i which gives v i = ϵ ijk ω j c k = Y ik c k . In matrix form Y = 0 ω 1 ω 2 ω 2 0 ω 3 ω 2 ω 3 0 We see that by choosing ω 1 = x 12 , ω 2 = x 13 , and ω 3 = x 23 we can set Y = X . With this choice of components ω i we have X ik = ϵ ijk ω j The equivalence between X and Y proves formula (4). Let us return to the issue of time-dependent e i ( t ). Formula ˙ c i = ϵ ijk ω j c k holds regardless time dependence of e i . Components ω i are given by the elements of matrix X that describes the rotation of the { e i } frame relative to the { e i } frame. Introduction of the vector ⃗ω = ω i e i is a convenience which allows one to write down formula (4). The latter formula is only true when the frame { e i } is at rest. But we can rewrite it in a form e i ˙ c i = [ ω j e j × e k c k ] , (5) which is true even if e i ( t ) are time-dependent. It’s just that in this case the l.h.s. is not equal to ˙
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notes_rotations_ver2011 - 5 5.1 Rotations Rotation matrix...

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