Ivancevic_Applied-Diff-Geom

Applied Jet Geometry 937Note that, if we consider another Lepagian equivalent of the Lagrangiandensity L, the SEM transformation law takes the forms ∗ L τ ΓL +ddx α [τ µ J ′ Γ α µ(s)] ω ≈ 0,where J ′ Γ α µ = J Γαµ − ddx ν [(∂ µs i − Γ i µ)c λνi ],that is, the SEM–tensors J ′ Γ α αµ and J Γ µ differ from each other in thesuperpotential–type term: − ddx[(∂ ν µ s i − Γ i µ)c λνi ].In particular, if the bundle Y has a fibre metric a Y ij , one can choosec µνi = a Y ijg µα g νβ R j αβ ,where R is the curvature of the connection Γ on the bundle Y and g is ametric on X. In this case, the superpotential contribution into the SEM–tensor is equal to − ddx[a Y ν ij gλα g νβ (∂ µ s i − Γ i µ)R j αβ ].Let us now consider the weak identity (5.408) when a vector–field τ onthe base X induces a vector–field on Y by means of different connectionsΓ and Γ ′ on Y −→ X. Their difference result in the weak identity[τ µ σ i µ∂ i + (∂ α (τ µ σ i µ) + y j α∂ j (τ µ σ i µ))∂ α i ]L − ̂∂ α [π α i τ µ σ i µ] ≈ 0 (5.411)where σ = Γ ′ − Γ is a soldering form on the bundle Y −→ X andτ⌋σ = τ µ σ i µ∂ i (5.412)is a vertical vector–field. It is clear that the identity (5.411) is exactly theweak identity (5.405) in case of the vertival vector–field (5.412).It follows that every SEM transformation law contains a Noether transformationlaw. Conversely, every Noether transformation law associatedwith a vertical vector–field u V on Y −→ X can be get as the difference of twoSEM transformation laws if the vector–field u V takes the form u V = τ⌋σ,where σ is some soldering form on Y and τ is a vector–field on X. Infield theory, this representation fails to be unique. On the contrary, inNewtonian mechanics there is the 1–1 correspondence between the verticalvector–fields and the soldering forms on the bundle R × F −→ F.Note that one can consider the pull–back of the first–order Lagrangiandensity L and their Lepagian equivalents onto the infinite order jet spaceJ ∞ Y . In this case, there exists the canonical lift τ ∞ H (5.59) of a vector–fieldτ on X onto J ∞ Y . One can treat this lift as the horizontal lift of τ by