Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 505Pendulum GeometryWe use the Riemannian manifold (R 2 , δ ij ). The small oscillations of a planependulum are described as solutions of the following differential systemgiving the plane pendulum flow,ẋ 1 = −x 2 , ẋ 2 = x 1 . (4.25)In this case, the set {x 1 (t) = 0, x 2 (t) = 0, (t ∈ R)} is the equilibriumpoint andx 1 (t) = c 1 cos t + c 2 sin t,x 2 (t) = c 1 sin t − c 2 cos tis the general solution, which is a family of circles with a common center.Let X = (X 1 , X 2 ), X 1 (x 1 , x 2 ) = −x 2 , X 2 (x 1 , x 2 ) = x 1 ,f(x 1 , x 2 ) = 1 2 (x2 1 + x 2 2), curl X = (0, 0, 2), div X = 0.The pendulum flow conserves the areas. The prolongation by derivationof the kinematic system (4.25) is [Udriste (2000)]ẍ i = ∂X iẋ j ,∂x j(i, j = 1, 2)or ẍ 1 = −ẋ 2 , ẍ 2 = ẋ 1 .This prolongation admits a family of circles as the general solutionx 1 (t) = a 1 cos t+a 2 sin t+h, x 2 (t) = a 1 sin t−a 2 cos t+k, (t ∈ R).The pendulum geometrodynamics is described byẍ i = ∂f ( ∂Xi+ − ∂X )jẋ j , (i, j = 1, 2),∂x i ∂x j ∂x ior ẍ 1 = x 1 − 2ẋ 2 , ẍ 2 = x 2 + 2ẋ 1 , (4.26)with a family of spirals as the general solutionx 1 (t) = b 1 cos t + b 2 sin t + b 3 t cos t + b 4 t sin t,x 2 (t) = b 1 sin t − b 2 cos t + b 3 t sin t − b 4 t cos t,(t ∈ R).