Variational Equations
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Variational Equations . δ r = δv . . δ v = ∇∇Uδ r
.
δ x = δV x .
δ y = δV y .
δ z = δV z ∂ 2U ∂ 2U ∂ 2U δVx = 2 δx + δy + δz ∂x ∂x∂y ∂x∂z
δV y =
∂ 2U ∂ 2U ∂ 2U δx + 2 δy + δz ∂y∂x ∂y ∂y∂z
δVz =
∂ 2U ∂ 2U ∂ 2U δx + δy + 2 δz ∂z∂x ∂z∂y ∂z 2
2
2
∂ 2U ∂ 2U ∂r ∂ 2U ∂φ ∂ 2U ∂λ ∂ 2U ∂r ∂φ ∂ 2U ∂r ∂λ ∂ 2U ∂φ ∂λ = 2 + +2 +2 + 2 +2 ∂r∂ϕ ∂x ∂x ∂r∂λ ∂x ∂x ∂λ∂ϕ ∂x ∂x ∂x 2 ∂r ∂x ∂ϕ 2 ∂x ∂λ ∂x ∂U ∂ 2 r ∂U ∂ 2φ ∂U ∂ 2 λ + + + ∂r ∂x 2 ∂φ ∂x 2 ∂λ ∂x 2 2
2
∂λ ∂ 2U ∂r ∂φ ∂ 2U ∂r ∂λ ∂ 2U ∂φ ∂λ + 2 +2 +2 ∂r∂ϕ ∂y ∂y ∂r∂λ ∂y ∂y ∂λ∂ϕ ∂y ∂y ∂y
2
∂ 2U ∂r ∂φ ∂ 2U ∂r ∂λ ∂ 2U ∂φ ∂λ ∂λ +2 +2 +2 ∂r∂ϕ ∂z ∂z ∂r∂λ ∂z ∂z ∂λ∂ϕ ∂z ∂z ∂z
∂r ∂ 2U ∂φ ∂ 2U + + ∂ϕ 2 ∂y ∂λ 2 ∂y ∂U ∂ 2 r ∂U ∂ 2φ ∂U ∂ 2 λ + + + ∂r ∂y 2 ∂φ ∂y 2 ∂λ ∂y 2 ∂ 2U ∂ 2U = 2 ∂y 2 ∂r
∂ 2U ∂ 2U = 2 ∂z 2 ∂r +
2
∂ 2U ∂r + ∂ϕ 2 ∂z
∂ 2U ∂φ + 2 ∂λ ∂z
2
2
∂U ∂ 2 r ∂U ∂ 2φ ∂U ∂ 2 λ + + ∂r ∂z 2 ∂φ ∂z 2 ∂λ ∂z 2
∂r ∂r ∂ 2U ∂φ ∂φ ∂ 2U ∂λ ∂λ ∂ 2U ∂φ ∂r ∂φ ∂r ∂ 2U ∂λ ∂r ∂λ ∂r + + 2 + + + + 2 ∂x ∂y ∂φ ∂x ∂y ∂λ ∂x ∂y ∂r∂φ ∂x ∂y ∂y ∂x ∂r∂λ ∂x ∂y ∂y ∂x ∂ 2U ∂λ ∂φ ∂λ ∂φ ∂U ∂ 2 r ∂U ∂ 2φ ∂U ∂ 2 λ + + + + + ∂λ∂φ ∂x ∂y ∂y ∂x ∂r ∂x∂y ∂φ ∂x∂y ∂λ ∂x∂y ∂ 2U ∂ 2U = ∂x∂y ∂r 2
∂ 2U ∂ 2U = ∂x∂z ∂r 2 +
2 ∂φ ∂φ ∂ U + 2 ∂x ∂z ∂λ
2 2 ∂λ ∂λ ∂ U ∂φ ∂r ∂φ ∂r ∂ U ∂λ ∂r ∂λ ∂r + + + + ∂x ∂z ∂r∂φ ∂x ∂z ∂z ∂x ∂r∂λ ∂x ∂z ∂z ∂x
∂ 2U ∂λ ∂φ ∂λ ∂φ ∂U ∂ 2 r ∂U ∂ 2φ ∂U ∂ 2 λ + + + + ∂λ∂φ ∂x ∂z ∂z ∂x ∂r ∂x∂z ∂φ ∂x∂z ∂λ ∂x∂z
∂ 2U ∂ 2U = ∂y∂z ∂r 2 +
2 ∂r ∂r ∂ U + 2 ∂x ∂z ∂φ
∂r ∂r ∂ 2U + 2 ∂y ∂z ∂φ
∂φ ∂φ ∂ 2U + 2 ∂y ∂z ∂λ
∂λ ∂λ ∂ 2U ∂φ ∂r ∂φ ∂r ∂ 2U ∂λ ∂r ∂λ ∂r + + + + ∂y ∂z ∂r∂φ ∂y ∂z ∂z ∂y ∂r∂λ ∂y ∂z ∂z ∂y
∂ 2U ∂λ ∂φ ∂λ ∂φ ∂U ∂ 2 r ∂U ∂ 2φ ∂U ∂ 2 λ + + + + ∂λ∂φ ∂y ∂z ∂z ∂y ∂r ∂y∂z ∂φ ∂y∂z ∂λ ∂y∂z
∂ 2U [ 2 ∂r
∂ 2U ∂φ 2
∂ 2U ∂λ2
∂ 2U ∂r∂φ
∂ 2U ∂r∂λ
∂ 2U ∂λ∂φ
∂U ∂r
∂U ∂φ
∂U ]* ∂λ
.
δ x = δV x .
δ y = δV y .
δ z = δV z ∂ 2U ∂ 2U ∂ 2U δVx = 2 δx + δy + δz ∂x ∂x∂y ∂x∂z
δV y =
∂ 2U ∂ 2U ∂ 2U δx + 2 δy + δz ∂y∂x ∂y ∂y∂z
δVz =
∂ 2U ∂ 2U ∂ 2U δx + δy + 2 δz ∂z∂x ∂z∂y ∂z 2
2
2
∂ 2U ∂ 2U ∂r ∂ 2U ∂φ ∂ 2U ∂λ ∂ 2U ∂r ∂φ ∂ 2U ∂r ∂λ ∂ 2U ∂φ ∂λ = 2 + +2 +2 + 2 +2 ∂r∂ϕ ∂x ∂x ∂r∂λ ∂x ∂x ∂λ∂ϕ ∂x ∂x ∂x 2 ∂r ∂x ∂ϕ 2 ∂x ∂λ ∂x ∂U ∂ 2 r ∂U ∂ 2φ ∂U ∂ 2 λ + + + ∂r ∂x 2 ∂φ ∂x 2 ∂λ ∂x 2 2
2
∂λ ∂ 2U ∂r ∂φ ∂ 2U ∂r ∂λ ∂ 2U ∂φ ∂λ + 2 +2 +2 ∂r∂ϕ ∂y ∂y ∂r∂λ ∂y ∂y ∂λ∂ϕ ∂y ∂y ∂y
2
∂ 2U ∂r ∂φ ∂ 2U ∂r ∂λ ∂ 2U ∂φ ∂λ ∂λ +2 +2 +2 ∂r∂ϕ ∂z ∂z ∂r∂λ ∂z ∂z ∂λ∂ϕ ∂z ∂z ∂z
∂r ∂ 2U ∂φ ∂ 2U + + ∂ϕ 2 ∂y ∂λ 2 ∂y ∂U ∂ 2 r ∂U ∂ 2φ ∂U ∂ 2 λ + + + ∂r ∂y 2 ∂φ ∂y 2 ∂λ ∂y 2 ∂ 2U ∂ 2U = 2 ∂y 2 ∂r
∂ 2U ∂ 2U = 2 ∂z 2 ∂r +
2
∂ 2U ∂r + ∂ϕ 2 ∂z
∂ 2U ∂φ + 2 ∂λ ∂z
2
2
∂U ∂ 2 r ∂U ∂ 2φ ∂U ∂ 2 λ + + ∂r ∂z 2 ∂φ ∂z 2 ∂λ ∂z 2
∂r ∂r ∂ 2U ∂φ ∂φ ∂ 2U ∂λ ∂λ ∂ 2U ∂φ ∂r ∂φ ∂r ∂ 2U ∂λ ∂r ∂λ ∂r + + 2 + + + + 2 ∂x ∂y ∂φ ∂x ∂y ∂λ ∂x ∂y ∂r∂φ ∂x ∂y ∂y ∂x ∂r∂λ ∂x ∂y ∂y ∂x ∂ 2U ∂λ ∂φ ∂λ ∂φ ∂U ∂ 2 r ∂U ∂ 2φ ∂U ∂ 2 λ + + + + + ∂λ∂φ ∂x ∂y ∂y ∂x ∂r ∂x∂y ∂φ ∂x∂y ∂λ ∂x∂y ∂ 2U ∂ 2U = ∂x∂y ∂r 2
∂ 2U ∂ 2U = ∂x∂z ∂r 2 +
2 ∂φ ∂φ ∂ U + 2 ∂x ∂z ∂λ
2 2 ∂λ ∂λ ∂ U ∂φ ∂r ∂φ ∂r ∂ U ∂λ ∂r ∂λ ∂r + + + + ∂x ∂z ∂r∂φ ∂x ∂z ∂z ∂x ∂r∂λ ∂x ∂z ∂z ∂x
∂ 2U ∂λ ∂φ ∂λ ∂φ ∂U ∂ 2 r ∂U ∂ 2φ ∂U ∂ 2 λ + + + + ∂λ∂φ ∂x ∂z ∂z ∂x ∂r ∂x∂z ∂φ ∂x∂z ∂λ ∂x∂z
∂ 2U ∂ 2U = ∂y∂z ∂r 2 +
2 ∂r ∂r ∂ U + 2 ∂x ∂z ∂φ
∂r ∂r ∂ 2U + 2 ∂y ∂z ∂φ
∂φ ∂φ ∂ 2U + 2 ∂y ∂z ∂λ
∂λ ∂λ ∂ 2U ∂φ ∂r ∂φ ∂r ∂ 2U ∂λ ∂r ∂λ ∂r + + + + ∂y ∂z ∂r∂φ ∂y ∂z ∂z ∂y ∂r∂λ ∂y ∂z ∂z ∂y
∂ 2U ∂λ ∂φ ∂λ ∂φ ∂U ∂ 2 r ∂U ∂ 2φ ∂U ∂ 2 λ + + + + ∂λ∂φ ∂y ∂z ∂z ∂y ∂r ∂y∂z ∂φ ∂y∂z ∂λ ∂y∂z
∂ 2U [ 2 ∂r
∂ 2U ∂φ 2
∂ 2U ∂λ2
∂ 2U ∂r∂φ
∂ 2U ∂r∂λ
∂ 2U ∂λ∂φ
∂U ∂r
∂U ∂φ
∂U ]* ∂λ
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