Ivancevic_Applied-Diff-Geom

Applied Jet Geometry 939This equality shows that ˜Γ is a Hamiltonian connection andH Γ = Γ⌋θ = p α i dy i ∧ ω α − p α i Γ i αωis a Hamiltonian form.Let H be a Hamiltonian form. For any exterior horizontal density ˜H =˜Hω on Π −→ X, the form H + ˜H is a Hamiltonian form. Conversely, ifH and H ′ are Hamiltonian forms, their difference H − H ′ is an exteriorhorizontal density on Π −→ X.Thus, Hamiltonian forms constitute an affine space modelled on a linearspace of the exterior horizontal densities on Π −→ X. It follows that everyHamiltonian form on Π can be given by the expression (5.396) where Γis some connection on Y −→ X. Moreover, a Hamiltonian form has thecanonical splitting (5.396) as follows.Every Hamiltonian form H implies the momentum mapĤ : Π −→ J 1 (X, Y ),y i α ◦ Ĥ = ∂i αH,and the associated connection Γ H = Ĥ ◦ ̂0 on Y where ̂0 is the global zerosection of Π → Y . As a consequence, we have the canonical splittingH = H ΓH − ˜H.The Hamiltonian operator E H of a Hamiltonian form H is defined to bethe first–order differential operator on Π → X,whereE H : j 1 Π → ∧ n+1 T ∗ Π,E H = dH − ̂Ω = [(y(λ) i − ∂i αH)dp α i − (p α iλ + ∂ i H)dy i ] ∧(5.413)ω̂Ω = dpαi ∧ dy i ∧ω α + p α iλdy i ∧ ω − y i (λ) dpα i ∧ ωis the pull–back of the multisymplectic form (5.394) onto j 1 Π.For any connection γ on Π → X, we haveE H ◦ γ = dH − γ⌋Ω.It follows that γ is a Hamiltonian connection for a Hamiltonian form H iffit takes its values into Ker E H given by the coordinate relationsy i (λ) = ∂i αH, p α iλ = −∂ i H. (5.414)Let a Hamiltonian connection γ has an integral section r of Π −→ X,that is, γ ◦ r = j 1 r. Then, the algebraic equations (5.414) are brought intothe first–order differential Hamiltonian equations (5.397).