Download Variations, Geometry and Physics: In Honour of Demeter by Olga Krupkova, David Saunders PDF
By Olga Krupkova, David Saunders
This booklet is a suite of survey articles in a wide box of the geometrical thought of the calculus of adaptations and its functions in research, geometry and physics. it's a commemorative quantity to have a good time the sixty-fifth birthday of Professor Krupa, one of many founders of recent geometric variational idea, and an enormous contributor to this subject and its functions over the last thirty-five years. the entire authors invited to give a contribution to this quantity have verified excessive reputations of their box. The publication will completely offer quite a few vital effects, strategies and purposes which are frequently on hand in basic terms by way of consulting unique papers in lots of varied journals. will probably be of curiosity to researchers in variational calculus, mathematical physics and the opposite similar parts of differential equations, typical operators and geometric buildings. additionally, it is going to turn into an enormous resource of present study for doctoral scholars and postdoctorals in those fields.
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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.