Ivancevic_Applied-Diff-Geom

938 Applied Differential Geometry: A Modern Introductionmeans of the canonical connection on the bundle J ∞ Y −→ X, given byΓ ∞ = dx µ ⊗(∂ µ + y i ∂ i + y i α∂ α i + · · · ).Multimomentum Hamiltonian FormalismLet Π be the Legendre bundle (5.393) coordinated by (x α , y i , p α i ). ByJ 1 (X, Π) is meant the first–order jet space of Π −→ X. It is coordinated by(x α , y i , p α i , yi (µ) , pα iµ ). The Legendre manifold Π carries the generalized Liouvilleformθ = −p α i dy i ∧ ω ⊗ ∂ αand the polysymplectic form Ω (5.394).The Hamiltonian formalism in fibred manifolds is formulated intrinsicallyin terms of Hamiltonian connections which play the role similar tothat of Hamiltonian vector–fields in the symplectic geometry [Sardanashvily(1993)].We say that a jet field (resp. a connection)γ = dx α ⊗ (∂ α + γ i (λ) ∂ i + γ µ iλ ∂i µ)on the Legendre manifold Π −→ X is a Hamiltonian jet field (resp.Hamiltonian connection) if the following exterior form is closed:aγ⌋Ω = dp α i ∧ dy i ∧ ω α + γ α iλdy i ∧ ω − γ i (λ) dpα i ∧ ω.An exterior n−form H on the Legendre manifold Π is called a Hamiltonianform if, on an open neighborhood of each point of Π, there existsa Hamiltonian jet–field satisfying the equation γ⌋Ω = dH, i.e., if there existsa Hamiltonian connection satisfying the equation (5.395). Hamiltonianconnections constitute an affine subspace of connections on Π → X. Thefollowing construction shows that this subspace is not empty.Every connection Γ on Y → X is lifted to the connectionγ = ˜Γ = dx α ⊗ [∂ α + Γ i α(y)∂ i + (−∂ j Γ i α(y)p µ i − Kµ νλ(x)p ν j + K α αλ(x)p µ j )∂j µ]on Π → X, whereK = dx α ⊗ (∂ α + K µ ∂νλẋ µ )∂ẋ νis a linear symmetric connection on T ∗ X. We have the equality˜Γ⌋Ω = d(Γ⌋θ).