Ivancevic_Applied-Diff-Geom

936 Applied Differential Geometry: A Modern IntroductionOne can simplify it as follows:τ µ {[∂ µ + Γ i µ∂ i + (∂ α Γ i µ + y j α∂ j Γ i µ)∂ α i ]L − ̂∂ α [π α i (Γ i µ − y i µ) + δ α µL] ≈ 0.Let us emphasize that this relation takes place for arbitrary vector–field τon X. Therefore, it is equivalent to the system of the weak identities[∂ µ + Γ i µ∂ i + (∂ α Γ i µ + y j α∂ j Γ i µ)∂ α i ]L − ̂∂ α [π α i (Γ i µ − y i µ) + δ α µL] ≈ 0. (5.409)On solutions s of the Euler–Lagrangian equations, the weak identity(5.408) becomes the weak transformation laws ∗ L τ ΓL +ddx α [τ µ J Γαµ (s)]ω ≈ 0and to the equivalent system of the weak transformation laws[∂ µ +Γ i µ∂ i +(∂ α Γ i µ+∂ α s j ∂ j Γ i µ)∂ α i ]L+ ddx α [πα i (∂ µ s i −Γ i µ)−δ α µL] ≈ 0 (5.410)where J Γα µ (s) is the SEM–tensor given by the components of theT ∗ X−valued (n − 1)−form on X,J Γ (s) = −(Γ⌋Ξ L ) ◦ s = [π α i (∂ µ s i − Γ i µ) − δ α µL]dx µ ⊗ ω α .It is clear that the first and the second terms in (5.410) taken separatelyfail to be well–behaved objects. Therefore, only their combination mayresult in the satisfactory transformation or conservation law.For example, let a Lagrangian density L depend on a background metricg on the base X. In this case, we have∂ µ L = −t α β√|g| Γβµα , where t α β = g αγ t γβis the metric SEM–tensor (by definition), while Γ β µα are the Christoffelsymbols of the metric g. Then, the weak transformation law (5.410) takesthe form−t α β√|g|Γβµα + [Γ i µ∂ i + (∂ α Γ i µ + ∂ α s j ∂ j Γ i µ)∂ α i ]L+ ddx α [πα i (∂ µ s i − Γ i µ) − δ α µL] ≈ 0,and, under suitable conditions of symmetries of the Lagrangian density L,it may become the covariant conservation law ∇ α t α β = 0 where ∇ α denotesthe covariant derivative relative to the connection Γ β µα.