Ivancevic_Applied-Diff-Geom

888 Applied Differential Geometry: A Modern IntroductionSubstitution of u (5.239) in the formula (5.216) leads to the first variationalformula in the presence of background fields:∂ α u α L + [u α ∂ α + u A ∂ A + u i ∂ i + (d α u A − y A µ ∂ α u µ )∂ α A(5.240)+(d α u i − y i µ∂ α u µ )∂ α i ]L = (u A − y A α u α )∂ A L + π α Ad α (u A − y A µ u µ )+(u i − yαu i α )δ i L − d α [π α i (u µ yµ i − u i ) − u α L]. (5.241)Then we have on the shell (5.220) the weak identity∂ α u α L + [u α ∂ α + u A ∂ A + u i ∂ i + (d α u A − y A µ ∂ α u µ )∂ α A + (d α u i − y i µ∂ α u µ )∂ α i ]L≈ (u A − y A α u α )∂ A L + π α Ad α (u A − y A µ u µ ) − d α [π α i (u µ y i µ − u i ) − u α L].If a total Lagrangian L is invariant under gauge transformations of Y tot , weget the weak identity(u A − y A µ u µ )∂ A L + π α Ad α (u A − y A µ u µ ) ≈ d α J α , (5.242)which is the transformation law of the symmetry current J in the presenceof background fields.5.9.1 Lagrangian Conservation LawsIn the first–order Lagrangian field theory, we have the following differentialtransformation and conservation laws on solutions s : X −→ Y (5.221) ofthe Euler–Lagrangian equations (5.220).Recall that given fibre coordinates (x α , y i ) of Y , the jet space J 1 (X, Y )is equipped with the adapted coordinates (x α , y i , y i α), while the first–orderLagrangian density on J 1 (X, Y ) is defined as the mapL : J 1 (X, Y ) → ∧ n T ∗ X,(n = dim X),L = L(x α , y i , y i α)ω, with ω = dx 1 ∧ ... ∧ dx n .The corresponding first–order Euler–Lagrangian equations for sections s :X −→ J 1 (X, Y ) of the jet bundle J 1 (X, Y ) → X read∂ α s i = s i α, ∂ i L − (∂ α + s j α∂ j + ∂ α s j α∂ α j )∂ α i L = 0. (5.243)We consider the Lie derivatives of Lagrangian densities in order to getdifferential conservation laws. Letu = u α (x)∂ α + u i (y)∂ i