Ivancevic_Applied-Diff-Geom

504 Applied Differential Geometry: A Modern Introductionand the Hamiltonian equations asẋ = X H (x).Let (M, g) be a semi–Riemann nD manifold. Let X be a C ∞ vector–field on M, and ω = g ◦ F the two–form associated to the tensor–fieldF = ∇X − g −1 ⊗ g(∇X) via the metric g.The tangent bundle is usually equipped with the Sasakian metric G,induced by g,G = g ij dx i ⊗ dx j + g ij δy i ⊗ δy j .If (x i , y i ) are the coordinates of the point (x, y) ∈ T M and Γ i jkare thecomponents of the connection induced by g ij , then we have the followingdual frames [Udriste (2000)]( δδx i =(dx j ,∂∂x i − Γh ijy j ∂∂y h ,)∂∂y i ⊂ X (T M),δy j = dy j + Γ j hk yk dx h ) ⊂ X ∗ (T M).The dynamical system (4.23) lifts to T M as a Hamiltonian dynamicalsystem with respect to theandHamiltonian H = 1 g(ẋ, ẋ) − f(x), and2symplectic two–form Ω 1 = g ij dx i ωδy j .This can be verified by putting η 1 = g ij y i dx j , and dη 1 = −Ω 1 .The dynamical system (4.24) lifts to T M as a Hamiltonian dynamicalsystem with respect to the above Hamiltonian function and the symplectictwo–formΩ 2 = 1 2 ω ijdx i ωdx j + g ij dx i ωδy j .This can be verified by putting η 2 = −g ij X i dx j +g ij y i dx j , and dη 2 = −Ω 2 .In the remainder of this subsection, we give three examples in Euclideanspaces, so we can put all indices down (still summing over repeated indices).