rewrite of Appendix A

Appendix A rewrite PhL 3.1.17 This has been installed on 3.1.7, do not edit here. Appendix A: Derivation of R(ξ) and Properties of Rotation Matrices Here we derive equations (14.4) through (14.6) for spherical coordinates. First, just for reference, here are the three active rotation matrices used below : Rx(θ) = Ry(θ) = Rz(θ) = . (A.1) These are called "active" since they rotate a vector forward (counterclockwise) relative to fixed axes by amount θ according to the right hand rule when the thumb is aligned with the axis in question. From the usual picture of spherical coordinates, (A.2) one can see (by staring hard enough) that = Rz(φ) Ry(θ) = R R ≡ Rz(φ) Ry(θ) = Rz(φ) Ry(θ) = R = Rz(φ) Ry(θ) = R // Ry(θ) does nothing here (A.3) which we rewrite as 1 = R e3 where 1 = e3 = 2 = R e1 2 = e1 = 3 = R e2 3 = e2 = . (A.4) We can repair the ordering of the basis vectors en on the right using R2 = (,,) as follows e3 = R2 e1 = R2 = = (,,) e1 = R2 e2 = e2 = R2 e3 = . (A.5) Maple tells us that R2-1 = R2T and det(R2) = 1, confirming that R2 is a rotation. Putting (A.5) into (A.4), 1 = R R2 e1 2 = R R2 e2 3 = R R2 e3 (A.6) or i = [R R2] ei . (A.7) Recalling (14.4), n = R(ξ)-1en n = 1,2,3 en = (R(ξ))-1nm m or n = [R(ξ)]nm em (14.4) we have therefore found the sought-after matrix R(ξ), [R(ξ)]-1 = R R2 . (A.8) Specific evaluation gives R = Rz(φ) Ry(θ) = = (A.9) and then [R(ξ)]-1 = R R2 = = (A.10) Since [R(ξ)]-1 = [R(ξ)]T we obtain R(ξ) by transposing the above matrix, so R(ξ) = . (A.11) From the right side of equation (14.4) (quoted above) we can write, i = R(ξ)ij ej = R(ξ)i1 e1 + R(ξ)i2 e2 + R(ξ)i3 e3 . (A.12) This can be written in the alternative notation of (1.1.32) as, = = R(ξ) (A.13a) or = = R(ξ) (A.13b) or = cosφsinθ + sinφsinθ + cosθ = cosφcosθ + sinφcosθ - sinθ = -sinφ + cosφ (A.13c) which are the well known expressions for the spherical unit vectors in terms of the Cartesian ones. The inverse of the last three equations can be obtained in the following manner (matrix from (A.10)) = [R(ξ)]-1 = (A.14a) or = [R(ξ)]-1 = (A.14b) or = cosφsinθ + cosφcosθ - sinφ = sinφsinθ + sinφcosθ + cosφ = cosθ - sinθ . (A.14c) Appendix E contains further information on spherical coordinates. Some Properties of Rotation Matrices A rotation matrix R is always real orthogonal which means R-1 = RT. It follows that RRT = 1 ΣkRik(RT)kj = δi'j ΣkRikRjk = δi,j RTR = 1 Σk(RT)ikRkj = δi'j ΣkRkiRkj = δi,j . (A.15) Also RRT = 1 det(RRT) = det(1) [det(R)]2 = 1 For a rotation matrix, det(R) = +1. (A.16) The determinant of R may be written using a standard expansion for the determinant of a matrix, 1 = det(R) = ΣijkεijkRi1Rj2Rk3 , (A.17) where the permutation tensor was described in (1.5.3). This can be generalized to read εabc = ΣijkεijkRiaRjbRkc for any a,b,c (A.18) where (A.17) is the particular case with abc = 123. We leave the proof of (A.18) as a Reader Exercise. Now apply ΣaRna to both sides and sum on a to get ΣaRna εabc = ΣaRna ΣijkεijkRiaRjbRkc = Σijkεijk [ ΣaRnaRia] RjbRkc = Σijkεijk [δn,i] RjbRkc // (A.15) = Σjk εnjkRjbRkc . We have just derived the following rarely-stated property of any 3x3 rotation matrix R : ΣaRnaεabc = ΣjkεnjkRjbRkc for any n,b,c . (A.19) Theorem: If A' = RA, B' =RB and C' = RC, then C = A x B C' = A' x B' . (A.20) Proof : C' = A' x B' C'n = Σjkεnjk A'jB'k (RC)n = Σjkεnjk (RA)j(RB)k (ΣaRnaCa) = Σjkεnjk(ΣbRjbAb)(ΣcRkcBc) = = Σbc [ ΣjkεnjkRjbRkc] AbBc = Σbc [ ΣaRnaεabc] AbBc // (A.19) = ΣaRna [ΣbcεabcAbBc ] = Σa Rna [A x B ]a . In vector notation we have just shown that RC = R(AxB). Apply R-1 from the left to conclude that C = A x B, QED. Run the steps in reverse to prove . This theorem confirms the intuitive fact that if A,B,C all transform as normal vectors under R, then if C = AxB in Frame S, then C' = A'xB' in rotated Frame S' . The fact that AB = A'B' is more obvious and requires only (A.15) : Σj(A')j(B')j = Σj(ΣbRjbAb)(ΣcRjcBc) = Σbc δbcAbBc = ΣbAbBb .