adding two vectors in spherical coordinates

Adding two vectors in spherical coordinates PhL 10.26.12 The transform is given by x = rsinθcosφ ρ2 = x2+ y2 = r2sin2θ ρ = r sinθ y = rsinθsinφ z = rcosθ r2 = x2+ y2+z2 y/x = tanφ The inverse transformation is given by r = [x2+ y2+z2]1/2 r ≥ 0 ρ = [x2+ y2]1/2 ρ ≥ 0 cosθ = z/r sinθ = ρ/r cosφ = x/(rsinθ) = x/ρ sinφ = y/(rsinθ) = y/ρ Now consider two vectors r1 = (r1, θ1, φ1)sph = (x1, y1, z1) r2 = (r2, θ2, φ2)sph = (x2, y2, z2) r = r1 + r2 = (x,y,z) = (r,θ,φ)sph Adding in Cartesian space, x = r1sinθ1cosφ1 + r2sinθ2cosφ2 y = r1sinθ1sinφ1 + r2sinθ2sinφ2 z = r1cosθ1 + r2cosθ2 Then the inverse transformation maybe be used to find (r,θ,φ) as follows. We first find from Maple, r2 = (x1+ x2)2+ (y1+ y2)2 + (z1+ z2)2 = r12 + r22 + 2r1r2sinθ1sinθ2cosφ1cosφ2 + 2r1r2sinθ1sinθ2sinφ1sinφ2 + 2r1r2cosθ1cosθ2 = r12 + r22 + 2r1r2sinθ1sinθ2 [cosφ1cosφ2 + sinφ1sinφ2] + 2r1r2cosθ1cosθ2 = r12 + r22 + 2r1r2sinθ1sinθ2cos(φ1-φ2) + 2r1r2cosθ1cosθ2 = r12 + r22 + 2r1r2{ cosθ1cosθ2 + sinθ1sinθ2cos(φ1-φ2) } = r12 + r22 + 2r1r2cosγ cosγ ≡ cosθ1cosθ2 + sinθ1sinθ2cos(φ1-φ2) where γ is then the angle between the two vectors. That is to say r1 r2 = r1r2 cosγ as again Maple confirms. These results are old friends but I don't know where to find these friends: Jackson, Sneddon, Goldstein, particle physics books, somewhere. Meanwhile we take a piece of the above to get ρ2 = (x1+x2)2+ (y1+ y2)2 = r12 + r22 + 2r1r2sinθ1sinθ2cosφ1cosφ2 + 2r1r2sinθ1sinθ2sinφ1sinφ2 - r12cosθ12 - r22cosθ22 = r12 + r22 + 2r1r2sinθ1sinθ2 [cosφ1cosφ2 + sinφ1sinφ2] - r12cosθ12 - r22cosθ22 = r12sin2θ1 + r22sin2θ2 + 2r1r2sinθ1sinθ2cos(φ1-φ2) So we summarize these two results r2 = r12 + r22 + 2r1r2cosγ cosγ ≡ sinθ1sinθ2cos(φ1-φ2) + cosθ1cosθ2 ρ2 = r12sin2θ1 + r22sin2θ2 + 2r1r2sinθ1sinθ2cos(φ1-φ2) The sum of the two vectors in sphericals is then given by r = [r12 + r22 + 2r1r2cosγ]1/2 r ≥ 0 ρ = [r12sin2θ1 + r22sin2θ2 + 2r1r2sinθ1sinθ2cos(φ1-φ2]1/2 ρ ≥ 0 cosθ = z/r sinθ = ρ/r cosφ = x/ρ sinφ = y/ρ x = r1sinθ1cosφ1 + r2sinθ2cosφ2 y = r1sinθ1sinφ1 + r2sinθ2sinφ2 z = r1cosθ1 + r2cosθ2 Now let's define α ≡ r1/r2 r2 = α r1 and then the above becomes r/r1 = [1 + α2 + 2αcosγ]1/2 r ≥ 0 ρ/r1 = [sin2θ1 +α2sin2θ2 + 2αsinθ1sinθ2cos(φ1-φ2]1/2 ρ ≥ 0 x/r1 = sinθ1cosφ1 +αsinθ2cosφ2 y/r1 = sinθ1sinφ1 + αsinθ2sinφ2 z/r1 = cosθ1 + αcosθ2 cosθ = z/r = (cosθ1 + αcosθ2)/ [1 + α2 + 2αcosγ]1/2 sinθ = ρ/r = [sin2θ1 +α2sin2θ2 + 2αsinθ1sinθ2cos(φ1-φ2)]1/2 / [1 + α2 + 2αcosγ]1/2 cosφ = x/ρ = (sinθ1cosφ1 +αsinθ2cosφ2)/ [sin2θ1 +α2sin2θ2 + 2αsinθ1sinθ2cos(φ1-φ2]1/2 sinφ = y/ρ = (sinθ1sinφ1 + αsinθ2sinφ2) / [sin2θ1 +α2sin2θ2 + 2αsinθ1sinθ2cos(φ1-φ2]1/2 So here is the answer to the question asked: Here is the r,θ,φ for the sum of two vectors. r = [r12 + r22 + 2r1r2cosγ]1/2 cosγ ≡ sinθ1sinθ2cos(φ1-φ2) + cosθ1cosθ2 cosθ = z/r = (cosθ1 + αcosθ2)/ [1 + α2 + 2αcosγ]1/2 sinθ = ρ/r = [sin2θ1 +α2sin2θ2 + 2αsinθ1sinθ2cos(φ1-φ2)]1/2 / [1 + α2 + 2αcosγ]1/2 cosφ = x/ρ = (sinθ1cosφ1 +αsinθ2cosφ2)/ [sin2θ1 +α2sin2θ2 + 2αsinθ1sinθ2cos(φ1-φ2)]1/2 sinφ = y/ρ = (sinθ1sinφ1 + αsinθ2sinφ2) / [sin2θ1 +α2sin2θ2 + 2αsinθ1sinθ2cos(φ1-φ2)]1/2 Small α approximation Now rewrite the above to make it suitable for expanding for small α, but don't assume small α. [sin2θ1 +α2sin2θ2 + 2αsinθ1sinθ2cos(φ1-φ2)]1/2 = sinθ1 [1 + 2α(sinθ2/sinθ1)cos(φ1-φ2) +α2(sinθ2/sinθ1)2]1/2 Case that α << 1. We then get cosθ = (cosθ1 + αcosθ2)/ [1 + α2 + 2αcosγ]1/2 ≈ (cosθ1 + αcosθ2)[1 - αcosγ] = cosθ1 + α [cosθ2 - cosθ1cosγ] ≈ cosθ1 + α [cosθ2 - cosθ1(sinθ1sinθ2cos(φ1-φ2) + cosθ1cosθ2)] ≈ cosθ1 + α [cosθ2 - cosθ1sinθ1sinθ2cos(φ1-φ2) - cos2θ1cosθ2)] ≈ cosθ1 + α [cosθ2(1-cos2θ1) - cosθ1sinθ1sinθ2cos(φ1-φ2) ] ≈ cosθ1 + α [cosθ2sin2θ1 - cosθ1sinθ1sinθ2cos(φ1-φ2) ] ≈ cosθ1 + α sinθ1 [cosθ2sinθ1 - cosθ1sinθ2cos(φ1-φ2) ] ≈ cosθ1 - α sinθ1 [cosθ1sinθ2cos(φ1-φ2) – cosθ2sinθ1 ] sinθ = sinθ1 [1 + 2α(sinθ2/sinθ1)cos(φ1-φ2) +α2(sinθ2/sinθ1)2]1/2 / [1 + α2 + 2αcosγ]1/2 ≈ sinθ1 [1 + 2α(sinθ2/sinθ1)cos(φ1-φ2)]1/2 / [1 + 2αcosγ]1/2 ≈ sinθ1 [1 + α (sinθ2/sinθ1) cos(φ1-φ2)] [1 – αcosγ] ≈ sinθ1 [ 1 + α { (sinθ2/sinθ1) cos(φ1-φ2) - cosγ } ≈ sinθ1 + α { sinθ2 cos(φ1-φ2) - sinθ1cosγ } ≈ sinθ1 + α { sinθ2 cos(φ1-φ2) - sinθ1(sinθ1sinθ2cos(φ1-φ2) + cosθ1cosθ2) } ≈ sinθ1 + α { sinθ2 cos(φ1-φ2) - (sin2θ1sinθ2cos(φ1-φ2) + sinθ1cosθ1cosθ2) } ≈ sinθ1 + α { sinθ2 cos(φ1-φ2) - sin2θ1sinθ2cos(φ1-φ2) - sinθ1cosθ1cosθ2 } ≈ sinθ1 + α { sinθ2 [cos(φ1-φ2) - sin2θ1cos(φ1-φ2)] - sinθ1cosθ1cosθ2 } ≈ sinθ1 + α { sinθ2 cos(φ1-φ2) [1 - sin2θ1] - sinθ1cosθ1cosθ2 } ≈ sinθ1 + α { sinθ2 cos(φ1-φ2) cos2θ1 - sinθ1cosθ1cosθ2 } ≈ sinθ1 + α cosθ1 [ sinθ2 cosθ1 cos(φ1-φ2) - sinθ1cosθ2 ] cosφ = (sinθ1cosφ1 +αsinθ2cosφ2)/ { sinθ1 [1 + 2α(sinθ2/sinθ1)cos(φ1-φ2) +α2(sinθ2/sinθ1)2]1/2} ≈ (sinθ1cosφ1 +αsinθ2cosφ2)/ { sinθ1 [1 + 2α(sinθ2/sinθ1)cos(φ1-φ2)]1/2} ≈ (cosφ1 +α(sinθ2/sinθ1)cosφ2)/ [1 + 2α(sinθ2/sinθ1)cos(φ1-φ2)]1/2 ≈ (cosφ1 +α(sinθ2/sinθ1)cosφ2) [1 – α(sinθ2/sinθ1)cos(φ1-φ2)] ≈ cosφ1 + α { (sinθ2/sinθ1)cosφ2 – cosφ1 (sinθ2/sinθ1)cos(φ1-φ2)} ≈ cosφ1 + α (sinθ2/sinθ1){cosφ2 – cosφ1 cos(φ1-φ2)} sinφ = (sinθ1sinφ1 +αsinθ2sinφ2)/ { sinθ1 [1 + 2α(sinθ2/sinθ1)cos(φ1-φ2) +α2(sinθ2/sinθ1)2]1/2} ≈ (sinθ1sinφ1 +αsinθ2sinφ2)/ { sinθ1 [1 + 2α(sinθ2/sinθ1)cos(φ1-φ2)]1/2} ≈ (sinφ1 +α(sinθ2/sinθ1)sinφ2)/ [1 + 2α(sinθ2/sinθ1)cos(φ1-φ2)]1/2 ≈ (sinφ1 +α(sinθ2/sinθ1)sinφ2) [1 – α(sinθ2/sinθ1)cos(φ1-φ2)] ≈ sinφ1 + α { (sinθ2/sinθ1)sinφ2 – sinφ1 (sinθ2/sinθ1)cos(φ1-φ2)} ≈ cosφ1 + α (sinθ2/sinθ1){sinφ2 – sinφ1 cos(φ1-φ2)} And don't forget r/r1 = [1 + α2 + 2αcosγ]1/2 ≈ [1 + 2αcosγ]1/2 ≈ [1 + αcosγ] Summary of the case α = (r1/r2) << 1: r ≈ r1(1 + αcosγ) cosθ ≈ cosθ1 - α sinθ1 [cosθ1sinθ2cos(φ1-φ2) – cosθ2sinθ1 ] sinθ ≈ sinθ1 + α cosθ1 [ sinθ2 cosθ1 cos(φ1-φ2) - sinθ1cosθ2 ] cosφ ≈ cosφ1 + α (sinθ2/sinθ1)[cosφ2 – cosφ1 cos(φ1-φ2)] sinφ ≈ cosφ1 + α (sinθ2/sinθ1)[sinφ2 – sinφ1 cos(φ1-φ2)] In the case of the sun-earth system we have MS = 1.99 x 1030 kg RS = 696,000 km ME = 5.97 x 1024 kg RE = 6371 km (8.25) MM = 7.35 x 1022kg RM = 1737 km TM = 27.3 days rS-E = 1.50 x 108 km // average of aphelion and perihelion and perigee e = .016 rM-E = 386,000 km // average of apogee and perigee tilt = 1.45o e = .05 So α = r1/r2 = RE/ rS-E = 6371 / 1.50 x 108 = 6.371 / 1.50 x 105 = 4.25 x 10-5 ~ 10-4 So in this case we essentially have r = r1.