Ivancevic_Applied-Diff-Geom

Applied Jet Geometry 895vector–field τ on X, its canonical lift ˜τ on T reads˜τ = τ α ∂ α + u Aβ α∂ β τ α ∂ A .Let a Lagrangian density L on the configuration space J 1 (X, T ) beinvariant under general covarian transformations. Then, it satisfies theequality (5.255) which takes the coordinate form∂ α (τ α L) + u Aβ α∂ β τ α ∂ A L + ̂∂ α (u Aβ α∂ β τ α )∂AL α − yα A ∂ β τ α ∂ β AL = 0. (5.257)Due to the arbitrariness of the functions τ α , the equality (5.257) is equivalentto the system of the equalities∂ α L = 0,δ β αL + u Aβ α∂ A L + ̂∂ α (u Aβ α)∂AL α − yα A ∂ β AL = 0, (5.258)u Aβ α∂AL α + u Aα α∂ β AL = 0. (5.259)Note that the equality (5.258) can be brought into the formδ β αL + u Aβ αδ A L + ̂∂ α (u Aβ α∂AL) α = yα A ∂ β AL, (5.260)where δ A L are the variational derivatives of the Lagrangian density L.Substituting the relations (5.260) and (5.259) into the weak identitywe get the conservation laŵ∂ α [(u Aβ α∂ β τ α − y A α τ α )∂ α AL + τ α L] ≈ 0,̂∂ α [−u Aα αδ A Lτ α − ̂∂ α (u Aα α∂ α ALτ α )] ≈ 0, (5.261)where the conserved current is reduced to the superpotential termQ eτ α = −u Aα αδ A Lτ α − ̂∂ α (u Aα α∂ α ALτ α ). (5.262)For general field models, we have the product T × Y of a bundle T →X of geometrical objects and some other bundle Y → X. The lift of avector–field τ on the base X onto the corresponding configuration spaceJ 1 (X, T ) × J 1 (X, Y ) readsτ = j 1 0˜τ + τ α Γ i α∂ i + (∂ α (τ α Γ i α) + τ α y j α∂ j Γ i α − y i α∂ α τ α )∂ α iwhere Γ is a connection on the fibre bundle Y → X.In this case, we cannot say anything about the general covariance conditionindependently on the invariance of a Lagrangian density with respectto the internal symmetries.