Ivancevic_Applied-Diff-Geom

856 Applied Differential Geometry: A Modern Introductionand plays the role of the homogeneous momentum phase–space of time–dependent mechanics. It admits the canonical Liouville form Ξ = p α dq α ,the canonical symplectic form Ω T = dΞ, and the corresponding Poissonbracket{f, f ′ } T = ∂ α f∂ α f ′ − ∂ α f∂ α f ′ , (f, f ′ ∈ C ∞ (T ∗ M)). (5.161)There is a canonical 1D fibre bundleζ : T ∗ M → V ∗ M, (5.162)whose kernel is the annihilator of the vertical tangent bundle V M ⊂ T M.The transformation law (5.160) shows that it is a trivial affine bundle.Indeed, given a global section h of ζ, one can equip T ∗ M with the fibrecoordinate r = p − h possessing the identity transition functions.The fibre bundle (5.162) gives the vertical cotangent bundle V ∗ M withthe canonical Poisson structure {, } V such thatζ ∗ {f, f ′ } V = {ζ ∗ f, ζ ∗ f ′ } T , (5.163){f, f ′ } V = ∂ k f∂ k f ′ − ∂ k f∂ k f ′ , (5.164)for all f, f ′ ∈ C ∞ (V ∗ M). The corresponding symplectic foliation coincideswith the fibration V ∗ M → R.However, the Poisson structure (5.164) fails to give any dynamical equationon the momentum phase–space V ∗ M because Hamiltonian vector–fieldsϑ f = ∂ k f∂ k − ∂ k f∂ k , ϑ f ⌋df ′ = {f, f ′ } V , (f, f ′ ∈ C ∞ (V ∗ M)),of functions on V ∗ M are vertical. Hamiltonian dynamics of time–dependent mechanics is described in a different way as a particular Hamiltoniandynamics on fibre bundles [Mangiarotti and Sardanashvily (1998);Giachetta et. al. (1997)].A Hamiltonian on the momentum phase–space V ∗ M → R of time–dependent mechanics is defined as a global sectionh : V ∗ M → T ∗ M, p ◦ h = −H(t, q j , p j ),of the affine bundle ζ (5.162). It induces the pull–back Hamiltonian formH = h ∗ Ξ = p k dq k − Hdt, (5.165)