Ivancevic_Applied-Diff-Geom

932 Applied Differential Geometry: A Modern Introductionfor some Hamiltonian connection γ. The key point lies in the fact thatevery Hamiltonian form admits the following splittingH = p α i dy i ∧ ω α − p α i Γ i αω − ˜H Γ ω = p α i dy i ∧ ω α − Hω, ω α = ∂ α ⌋ω,(5.396)where Γ is a connection on Y → X.Given the splitting (5.396), the equality (5.395) becomes the Hamiltonianequations∂ α r i = ∂ i αH, ∂ α r α i = −∂ i H (5.397)for sections r of Π −→ X.The Hamiltonian equations (5.397) are the multimomentum generalizationof the standard Hamiltonian equations in mechanics. The correspondingmultimomentum generalization of the conventional energy conservationlaw (5.384) is the weak identityτ µ [(∂ µ + Γ i µ∂ i − ∂ i Γ j µrj α ∂λ) i ˜H Γ −ddx α T Γ α µ(r)] ≈ τ µ ri α Rλµ, i (5.398)αT Γ µ (r) = [ri α ∂µ i ˜H Γ − δ α µ(ri α ∂α i ˜H Γ − ˜H Γ )], (5.399)whereR i λµ = ∂ α Γ i µ − ∂ µ Γ i α + Γ j α∂ j Γ i µ − Γ j µ∂ j Γ i αis the curvature of the connection Γ. One can think of the tensor (5.399)as being the Hamiltonian SEM–tensor.If a Lagrangian density is regular, the multimomentum Hamiltonianformalism is equivalent to the Lagrangian formalism, otherwise in caseof degenerate Lagrangian densities. In field theory, if a Lagrangian densityis not regular, the Euler–Lagrangian equations become underdeterminedand require supplementary gauge–type conditions. In gauge theory,they are the familiar gauge conditions. However, in general case,the gauge–type conditions remain elusive. In the framework of themultimomentum Hamiltonian formalism, they appear automatically asa part of the Hamiltonian equations. The key point consists in thefact that, given a degenerate Lagrangian density, one must consider afamily of different associated Hamiltonian forms in order to exhaust allsolutions of the Euler–Lagrangian equations [Cariñena et. al. (1991);Sardanashvily (1993)].Lagrangian densities of all realistic field models are quadratic or affinein the velocity coordinates yµ. i Complete family of Hamiltonian forms associatedwith such a Lagrangian density always exists [Sardanashvily (1993);Sardanashvily (1994)]. Moreover, these Hamiltonian forms differ from each