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Next we put ω0 = dx1 ∧ dx2 ∧ ... 3) if dim X = 1, the operators of total derivative. A mapping γ : U → Y defined on an open subset U ⊂ X is called a section of the fibred manifold π, if π ◦ γ = idU . , the r-jet prolongation of a section γ of π. ξ = 0. Quite similarly one can define a πr,s -projectable or a πr,s -vertical vector field on J r Y , where r > s. Local flows of projectable vector fields transfer sections into sections; consequently, π-projectable vector fields on Y can be naturally prolonged to vector fields on J r Y as follows.
E. 9. Let dim X = 1. To every Lagrangian λ on J r Y there exists a unique Lepage equivalent. In fibred coordinates where λ = Ldt, it reads r−1 ∂L ∂L d ∂L d2 ∂L r−1 d − + − · · · + (−1) ωσ ∂q1σ dt ∂q2σ dt2 ∂q3σ dtr−1 ∂qrσ d ∂L σ ∂L σ ∂L − ωr−2 + σ ωr−1 . 19) The above form Θλ is called (higher order) Cartan form. For a first order Lagrangian the formula is reduced to the well-known one, introduced to the calculus of variations and classical mechanics by Whittaker [78] and Cartan [5]: Θλ = Ldt + ∂L σ ∂L ∂L ω = L − σ q˙σ dt + σ dq σ .
22-nd Winter School Geometry nad Physics, Srn´ı, Czech Republic, 2002) 185–190. [40] J. Sniatycki, On the geometric structure of classical field theory in Lagrangian formulation, Proc. Camb. Phil. Soc. 68 (1970) 475–484. ˇ anek, A representation of the variational sequence in higher order mechanics, [41] J. Stef´ In: Differential Geometry and its Applications (Proc. , Brno, Czech Republic, 1995, (J. Janyˇska, I. Kol´aˇr and J. ) Masaryk University, Brno, 1996) 469– 478. [42] F. Takens, A global version of the inverse problem of calculus of variations, J.