Ivancevic_Applied-Diff-Geom

900 Applied Differential Geometry: A Modern IntroductiondensityL = Lω : J 1 (X, Y ) → ∧ n T ∗ X, (ω = dx 1 ∧ · · · dx n , n = dim X),(5.269)on the jet space J 1 (X, Y ). The corresponding Euler–Lagrangian equations(∂ i − d α ∂ α i )L = 0, d α = ∂ α + y i α∂ i + y i λµ∂ µ i , (5.270)represent the subset of the 2–jet space J 2 (X, Y ) of Y , coordinated by(x µ , y i , yα, i yλµ i ). A section s of Y → X is a solution of these equationsif its second jet prolongation j 2 s lives in the subset (5.270).The phase–space of covariant (polysymplectic) Hamiltonian field theoryon a fibre bundle Y −→ X is the Legendre bundle (see (5.225) above)Π = ∧ n T ∗ X ⊗ V ∗ Y ⊗ T X = V ∗ Y ∧(∧ n−1 T ∗ X), (5.271)where V ∗ Y is the vertical cotangent bundle of Y → X. The Legendrebundle Π is equipped with the holonomic bundle coordinates (x α , y i , p µ i )compatible with the composite fibrationΠ π Y−→ Yadmitting the canonical polysymplectic formΩ = dp α i ∧ dy i ∧ ω ⊗ ∂ α .π−→ X, (5.272)A covariant Hamiltonian H on Π (5.272) is defined as a section p = −H ofthe trivial line bundle (i.e., 1D fibre bundle)Z Y = T ∗ Y ∧ (∧ n−1 T ∗ X) → Π, (5.273)equipped with holonomic bundle coordinates (x α , y i , p µ i , p). This fibre bundleadmits the canonical multisymplectic Liouville formΞ = pω + p α i dy i ∧ ω α , with ω α = ∂ α ⌋ω.The pull–back of Ξ onto Π by a Hamiltonian H gives the Hamiltonian formH = H ∗ Ξ Y = p α i dy i ∧ ω α − Hω (5.274)on Π. The corresponding covariant Hamiltonian equations on Π,y i α = ∂ i αH, p α λi = −∂ i H, (5.275)