Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 507Using L = 1 23∑(ẋ i ) 2 −i=13∑X i ẋ i + f, H = 1 2g ij = (H + f)δ ij , N j i = −F j i = −δ ih F jh , F ij = ∂X j∂x ii=13∑(ẋ i ) 2 − f,i=1− ∂X i∂x j,(i, j, h = 1, 2, 3), the solutions of the differential system (4.27) are horizontalpregeodesics of the Riemann–Jacobi–Lagrangian manifold (R 3 \E, g ij , N j i ), where E is the set of equilibrium points.Geometry of the ABC FlowWe use the Riemannian manifold (R 3 , δ ij ). One example of a fluid velocitythat contains exponential stretching and hence instability is the ABC flow,named after Arnold, Beltrami and Childress,ẋ 1 = A sin x 3 +C cos x 2 , ẋ 2 = B sin x 1 +A cos x 3 , ẋ 3 = C sin x 2 +B cos x 1 .For nonzero values of the constants A, B, C the preceding system is notglobally integrable. The topology of the flow lines is very complicated andcan only be investigated numerically to reveal regions of chaotic behavior.The ABC flow conserves the volumes since the ABC field is solenoidal.The ABC geometrodynamics is described by [Udriste (2000)]ẍ i = ∂f ( ∂Xi+ − ∂X )jẋ j , (i, j = 1, 2, 3).∂x i ∂x j ∂x iSince f = 1 2 (A + B + C + 2AC sin x 3 cos x 2 + 2BA sin x 1 cos x 3 +2CB sin x 2 cos x 1 ), and curl X = X, the ABC geometrodynamics is givenby the system,ẍ 1 = AB cos x 1 cos x 3 − BC sin x 1 sin x 2− (B cos x 1 + C sin x 2 )ẋ 2 + (B sin x 1 + A cos x 3 )ẋ 3 ,ẍ 2 = −AC sin x 2 sin x 3 + BC cos x 1 cos x 2+ (B cos x 1 + C sin x 2 )ẋ 1 − (A sin x 3 + C cos x 2 )ẋ 2 ,ẍ 3 = AC cos x 3 cos x 2 − BA sin x 1 sin x 3− (B sin x 1 + A cos x 3 )ẋ 1 + (C cos x 2 + A sin x 3 )ẋ 2 .