Ivancevic_Applied-Diff-Geom

908 Applied Differential Geometry: A Modern IntroductionThe bracket {H, f} V in this expression is not globally defined because H isnot a function on V ∗ Q. Therefore, the evolution operator (5.296) does notreduce to the Poisson bracket (5.293).Let us now consider the pull–back ζ ∗ H of the Hamiltonian form H(5.294) onto T ∗ Y . Then the differenceH ∗ = Ξ − ζ ∗ H = (p + H)dx (5.297)is a horizontal density on the fibre bundle T ∗ Y → X. It is a multisymplecticHamiltonian form. The corresponding Hamiltonian connection γ on T ∗ Y →X is given by the conditionwhere the mapγ(Ω) = dH ∗ , (5.298)γ(Ω) = dx ∧ [(∂ x + γ p ∂ p + γ i ∂ i + γ i ∂ i )⌋Ω]is induced by an endomorphism of T ∗ Y determined by the tangent–valuedform γ. We getγ = dx ⊗ (∂ x + γ p ∂ p + ∂ i H∂ i − ∂ i H∂ i ), (5.299)where the coefficient γ p is arbitrary. Note that this connection projects tothe connection γ H (5.295) on V ∗ Y → X. As a consequence, it defines theevolution operator whose restriction to the pull–back of functions on V ∗ Qis exactly the evolution operator (5.296). But now this operator locallyreduces to the Poisson bracket on T ∗ Y ,d γ f = {p + H, f}dx, (f ∈ C ∞ (V ∗ Y )). (5.300)However, this bracket is not globally defined, too, since p+H is a horizontaldensity, but not a function on T ∗ Y .Let us introduce the function E = ρ −1 (p + H) on T ∗ Y , where ρdx issome nowhere vanishing density on X. The Hamiltonian vector–field of Ewith respect to the symplectic form Ω on T ∗ Y readsϑ E = ρ −1 ∂ x − ∂ x E∂ p + ∂ i E∂ i − ∂ i E∂ i .This vector–field is horizontal with respect to the connection (5.299), whereγ p = −ρ∂ x E, and it determines this connection in the formγ = dx ⊗ (∂ x − ρ∂ x E∂ p + ρ∂ i E∂ i − ρ∂ i E∂ i ).