Ivancevic_Applied-Diff-Geom

Applied Jet Geometry 907First of all, let us study the case of fibre bundles Y −→ X over a 1Dorientable connected manifold X (i.e., X is either R or S 1 ). In this case, theLegendre bundle Π (5.283) is isomorphic to the vertical cotangent bundleV ∗ Y of Y −→ X coordinated by (x, y i , p i ), and the polysymplectic form Ω Π(5.284) on V ∗ Y readsΩ Π = dp i ∧ dy i ∧ dx ⊗ ∂ x . (5.291)Therefore, the homogeneous Legendre bundle (6.212) is the cotangent bundleT ∗ Y , coordinated by (x, y i , p, p i ), and the multisymplectic form (5.286)becomes the canonical symplectic form on T ∗ Y , given byΩ = dp ∧ dx + dp i ∧ dy i . (5.292)The vertical cotangent bundle V ∗ Ybracketadmits the canonical Poisson{f, f ′ } V = ∂ i f∂ i f ′ − ∂ i f∂ i f ′ , (f, f ′ ∈ C ∞ (V ∗ Y )), (5.293)given by the relationζ ∗ {f, f ′ } V = {ζ ∗ f, ζ ∗ f ′ },where {, } is the canonical Poisson bracket on T ∗ Y [Mangiarotti and Sardanashvily(1998); Sardanashvily (1998)]. However, the Poisson structure(5.293) fails to determine Hamiltonian dynamics on the fibre bundleV ∗ Q → X because all Hamiltonian vector–fields with respect to this structureare vertical. At the same time, in accordance with general polysymplecticformalism [Giachetta et. al. (1997)], a section h, p ◦ h = −H, of thefibre bundle V ∗ Y −→ T ∗ Y induces a polysymplectic Hamiltonian form onV ∗ Y ,H = p i dy i − Hdx. (5.294)The associated Hamiltonian connection on V ∗ Y −→ X with respect to thepolysymplectic form (5.291) isγ H = dx ⊗ (∂ x + ∂ i H∂ i − ∂ i H∂ i ). (5.295)It defines the Hamiltonian equation on V ∗ Y ,y i x = ∂ i H, p ix = −∂ i H.The corresponding evolution operator (5.290) takes the local formd γ f = (∂ x f + {H, f} V )dx, (f ∈ C ∞ (V ∗ Q)). (5.296)