By Olga Krupkova, David Saunders

This ebook is a suite of survey articles in a vast box of the geometrical conception of the calculus of diversifications 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 contemporary geometric variational conception, and an incredible contributor to this subject and its functions during the last thirty-five years. all of the authors invited to give a contribution to this quantity have tested excessive reputations of their box. The booklet will solely supply various very important effects, recommendations and purposes which are often on hand purely by means of consulting unique papers in lots of diverse journals. it is going to be of curiosity to researchers in variational calculus, mathematical physics and the opposite similar parts of differential equations, traditional operators and geometric buildings. additionally, it's going to develop into an immense resource of present study for doctoral scholars and postdoctorals in those fields.

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M. Vinogradov, Geometry of Jet Spaces and Nonlinear Partial Differential Equations (Advanced Studies in Contemporary Mathematics 1, Gordon & Breach, New York, 1986). [12] M. Krbek, J. Musilov´a and J. Kaˇsparov´a, The variational sequence: Local and global properties, In: Proceedings of the Seminar on Differential Geometry (Math. Publications 2, Silesian University, Opava, 2000) 15–38. [13] M. Krbek, J. Musilov´a and J. Kaˇsparov´a, The representation of the variational sequence in field theories, In: Steps in Differential Geometry (Proc.

In Legendre coordinates, De Donder–Hamilton equations read ∂y σ ∂H = j, j ∂x ∂pσ ∂pjσ ∂H = − σ. 40) A higher-order version of this result was first considered by De Donder [11]. However, the generalisation is not so straightforward, since Θλ may be not globally well-defined, and if “globalised”, is non-unique. Saving the property of being determined completely by the Lagrangian, one has to resign on global Hamilton equations. 41) and are defined on the domain W of the coordinates (xi , y σ , yjσ1 , yjσ1 j2 , .

1. Lepage Forms and the First Variation Variations Let us start with Lagrangians and variations, following [36]. Consider a fibred manifold π : Y → X, where X is an orientable manifold of dimension n. A Lagrangian of order r for π is defined to be a horizontal n-form λ on J r Y . , 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.