Ivancevic_Applied-Diff-Geom

Applied Jet Geometry 931Moreover, the covariant conservation law (5.391) fails to take place inthe affine–metric gravitation theory and in the gauge gravitation theory,e.g., in the presence of fermion fields.Thus, we have not any conventional energy–momentum conservationlaw in Lagrangian field theory. In particular, one may take different SEM–tensors for different field models and, moreover, different SEM–tensors fordifferent solutions of the same field equations. Just the latter in fact isthe above-mentioned symmetrization of the canonical energy–momentumtensor in gauge theory.Gauge theory exemplifies constraint field theories. Contemporary fieldmodels are almost always the constraint ones. To describe them, let us turnto the Hamiltonian formalism.When applied to field theory, the conventional Hamiltonian formalismtakes the form of the instantaneous Hamiltonian formalism where canonicalvariables are field functions at a given instant of time. The correspondingphase–space is infinite–dimensional, so that the Hamiltonian equations inthe bracket form fail to be differential equations.The true partners of the Lagrangian formalism in classical field theoryare polysymplectic and multisymplectic Hamiltonian machineries wherecanonical momenta correspond to derivatives of fields with respect to allworld coordinates, not only the temporal one [Cariñena et. al. (1991);Sardanashvily (1993)]. We here follow the multimomentum Hamiltonianformulation of field theory when the phase–space of fields is the Legendrebundle over YΠ = ∧ n T ∗ X ⊗ T X ⊗ V ∗ Y, (5.393)which is coordinated by (x α , y i , p α i ) [Sardanashvily (1993); Sardanashvily(1994)]. Every Lagrangian density L on J 1 (X, Y ) implies the Legendremorphism̂L : J 1 (X, Y ) −→ Π, p µ i ◦ ̂L = π µ i .The Legendre bundle (5.393) carries the polysymplectic formΩ = dp α i ∧ dy i ∧ ω ⊗ ∂ α . (5.394)Recall that one says that a connection γ on the fibred Legendre manifoldΠ → X is a Hamiltonian connection if the form γ⌋Ω is closed. Then, aHamiltonian form H on Π is defined to be an exterior form such thatdH = γ⌋Ω (5.395)