Ivancevic_Applied-Diff-Geom

884 Applied Differential Geometry: A Modern Introductionbe given byL j 1 uL = [∂ α u α L + (u α ∂ α + u i ∂ i + (d α u i − y i µ∂ α u µ )∂ α i )L] ω. (5.215)The first variational formula (5.109) gives its canonical decomposition (inaccordance with the general variational problem), which readsL j 1 uL = u V ⌋E L + d H h 0 (u⌋H L ) (5.216)= (u i − y i µu µ )(∂ i − d α ∂ α i )Lω − d α [π α i (u µ y i µ − u i ) − u α L]ω.In the canonical decomposition (5.216), u V = (u⌋θ i )∂ i ; the mapE L : J 2 (X, Y ) → T ∗ Y ∧ (∧ n T ∗ X), given by E L = (∂ i L − d α π α i )θ i ∧ ω,(5.217)(with π α i = ∂i α L) is the Euler–Lagrangian operator associated to the LagrangianL; and the mapH L : J 1 (X, Y ) → M Y = T ∗ Y ∧ (∧ n−1 T ∗ X), given by (5.218)H L = L + π α i θ i ∧ ω α = π α i dy i ∧ ω α + (L − π α i y i α)ω, (5.219)is called the Poincaré–Cartan form.The kernel of the Euler–Lagrangian operator E L (5.217) defines thesystem of second–order Euler–Lagrangian equations, in local coordinatesgiven by(∂ i − d α ∂ α i )L = 0, (5.220)A solution of these equations is a section s : X −→ Y of the fibre bundle Y−→ X, whose second–order jet prolongation j 2 s lives in (5.220), i.e.,∂ i L ◦ s − (∂ α + ∂ α s j ∂ j + ∂ α ∂ µ s j ∂ µ j )∂α i L ◦ s = 0. (5.221)Different Lagrangians L and L ′ can lead to the same Euler–Lagrangianoperator E L if their difference L 0 = L − L ′ is a variationally trivial Lagrangian,whose Euler–Lagrangian operator vanishes identically. A LagrangianL 0 is called variationally trivial iffL 0 = h 0 (ϕ), (5.222)where ϕ is a closed n−-form on Y . We have at least locally ϕ = dξ, andthenL 0 = h 0 (dξ) = d H (h 0 (ξ) = d α h 0 (ξ) α ω, h 0 (ξ) = h 0 (ξ) α ω α .