By J. Aguade, R. Kane
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Extra resources for Algebraic Topology, Barcelona 1986
24) u = u λ (x μ )∂λ + u i (x μ , y j )∂i be a projectable vector field on a fibre bundle Y → X . 7) into the horizontal and vertical parts over J 1 Y reads u = u H + u V = u λ (∂λ + yλi ∂i ) + (u i ∂i − yλi u λ ∂i ). 24) onto J 1 Y . 12). 26) λ − u ) − u L ]. 24). 28) reads Ju = Juλ ωλ = [πiλ (u μ yμi − u i ) − u λ L ]ωλ . 28) in a d H -exact term μλ ψ = dμ (ψi (u i − yνi u ν ))ωλ . 28) is linear in a vector field u. 27) associated to different symmetries u. For instance, let v = v i ∂i be a vertical vector field on Y → X .
In a general setting, one also considers other Lepage equivalents of L [91, 92]. 12) takes its values into a subbundle n−1 J 1 Y ×(T ∗ Y ∧( ∧ T ∗ X )) Y Y n of ∧ T ∗ J 1 Y . 14) is an imbedded subbundle i L : Z L → Z Y of the fibre bundle Z Y → Y . This morphism is called the homogeneous Legendre map. 13) is said to be the homogeneous Legendre bundle. 15) ∂y j ∂ y i μ p . 13) reads ( piμ , p) ◦ HL = (πiμ , L − yμi πiμ ). 8) modelled over the pull-back vector bundle n Π × ∧ T ∗ X → Π. 5) is exactly the composition of morphisms L = π Z Π ◦ HL : J 1 Y → Π.
I) Let E → X be another vector bundle and ζ a linear E(X )-valued differential operator on a C ∞ (X )-module E (X ) of sections of E → X . Then u ζ (ξ ) = (ζ ◦ ζ )(ξ ) also is a gauge symmetry of L parameterized by sections ξ of E → X . 13). 8). 6) associates to a gauge symmetry of a Lagrangian L the Noether identities (NI) of its Euler–Lagrange operator. 17). 6 for the notation). 17) are the NI for the Euler–Lagrange operator δL. 17) take a form iνμ u ia Ei − dμ (u iμ a Ei ) + dνμ (u a Ei ) = 0.