spherical coordinate motion example of generalized coordinates

(Example)

As an example let us get the equations in spherical coordinates for the motion.

Where

x = r cos𝜃, y = rsin𝜃 cosϕ, z = rsin 𝜃sinϕ,

[ ] m- 2 2 ˙2 2 2 ˙2 T = 2 ˙r + r 𝜃 r sin 𝜃ϕ .

∂T- = m ˙r, ∂r˙

[ ] ∂T- ˙2 2 ˙2 ∂r = mr 𝜃 + sin 𝜃ϕ ,

∂T- = mr2 ˙𝜃, ∂𝜃˙

∂T-= mr2 sin 𝜃 cos𝜃ϕ˙2, ∂𝜃

∂T- 2 2 ˙ ˙ = mr sin 𝜃 ϕ. ∂ϕ

[ ( ) ] ˙2 2 ˙2 δrW = m ¨r − r 𝜃 + sin 𝜃ϕ δr = R δr,

[ ( ) ] δ W = m d- r2𝜃˙ − r2sin𝜃 cos𝜃ϕ˙2 δ𝜃 = Θr δ𝜃, 𝜃 dt

-d ( 2 2 ˙) δϕW = m dt r sin 𝜃ϕ δϕ = Φr sin𝜃 δϕ;

{ [ ] } d2r ( d𝜃)2 ( dϕ )2 m --2-− r --- + sin2 𝜃 --- = R, dt dt dt

[ ( ) ( ) ] m- -d 2d𝜃- 2 d-ϕ 2 r dt r dt − r sin 𝜃 cos𝜃 dt = Θ,

( ) m d 2 2 dϕ rsin𝜃-dt r sin 𝜃dt- = Φ.

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