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L = L(xi , y σ , yjσ1 , . . jr ). e. a compact n-dimensional submanifold of X with boundary, and ΓΩ (π) the set of smooth sections of π over Ω. 1) is called the variational function or action function of the Lagrangian λ over Ω. Since J r γ ∗ η = 0 for any contact n-form η on J r Y , the action function remains the same if one considers the form λ + η instead of λ. In other words, for any n form ρ such that hρ = λ, J r γ∗λ = Ω J r γ ∗ ρ. 2) Ω Let γ : U → Y be a section defined on an open set U ⊂ X.

Using the Poincar´e lemma we get around every point x ∈ X an (n − 2)-form ϕ for which dϕ = f i ωi is a conserved current on X, so that the corresponding conservation law d(f i ωi ) = 0 takes a “divergence form” div f = 0. In mechanics (dim X = 1) the situation is simpler: a “conserved current” is a function, F , and a conservation law reads F ◦ J 2r−1 γ = const. Therefore, F is called constant of the motion. Since in field theory the Lepage equivalent of λ is not unique, there arises a question on how a conserved current depends upon a choice of a Lepage equivalent ρ of λ.

Sci. Nat. Univ. Purk. Brunensis XIV (10) (1973) pp. 65. [17] D. Krupka, On the structure of Euler-Lagrange mapping, Arch. Math. (Brno) 10 (1974) 353–358. [18] D. Krupka, A map associated to the Lepagean forms of the calculus of variations in fibered manifolds, Czechoslovak Math. J. 27 (1977) 114–118. [19] D. Krupka, Natural Lagrangian Structures, In: Differential geometry (Semester in Differential Geometry, Banach Center, Warsaw, 1979, Banach Center Publications 12, 1984) 185–210. [20] D. Krupka, On the local structure of Euler-Lagrange mapping of the calculus of variations, In: Differential Geometry and its Applications (Proc.

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Algorithmic Geometry [auth. unkn.]


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