Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 503we shall show that the prolongation (4.20) describes a conservative dynamicsof the vector–field X or of a particle which is sensitive to the vector–fieldX. The physical phenomenon produced by (4.21) or (4.22) was not yetstudied [Udriste (2000)].In the case F = 0, the kinematic system (4.17) prolongs to a potentialdynamical system with n degrees of freedom, namely∇= grad f. (4.23)dtẋIn the case F ≠ 0, the kinematic system (4.17) prolongs to a non–potential dynamical system with n degrees of freedom, namely∇= grad f + F (ẋ) . (4.24)dtẋLet us show that the dynamical systems (4.23) and (4.24) are conservative.To simplify the exposition we identity the tangent bundle T Mwith the cotangent bundle T ∗ M using the semi–Riemann metric g [Udriste(2000)]. The trajectories of the dynamical system (4.23) are the extremalsof the LagrangianL = 1 g (ẋ, ẋ) + f(x).2The trajectories of the dynamical system (4.24) are the extremals of theLagrangianL = 1 2 g (ẋ − X, ẋ − X) = 1 g (ẋ, ẋ) − g (X, ẋ) + f(x).2The dynamical systems (4.23) and (4.24) are conservative, the Hamiltonianbeing the same for both cases, namelyH = 1 g (ẋ, ẋ) − f(x).2The restriction of the Hamiltonian H to the flow of the vector–field X iszero.4.4.2 Hamiltonian Structures on the Tangent BundleLet (N, ω) be a 2nD symplectic phase–space manifold, and H : N → R bea C ∞ real function. We define the Hamiltonian gradient X H as being thevector–field which satisfiesω p (X H (x), v) = dH(x)(v),(for all v ∈ T x N),