Poincaré-Cartan forms in higher order variational calculus on fibred manifolds
type: Item
instance of: scholarly article
title: Poincaré-Cartan forms in higher order variational calculus on fibred manifolds
zbMATH Open document ID: 0606.49026
DOI: 10.4171/RMI/20
author: Jaime Muñoz Masqué
published in: Revista Matemática Iberoamericana
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/39833
review text: The paper provides a self-contained and clear introduction into the geometric theory of higher order multiple integral variational problems on a fibred manifold p:Y\(\to X\). The main part is devoted to a very thorough and explicit construction of the Poincaré-Cartan form \(\Theta\) that can be associated to an r-order Lagrange density Lv. A derivation \(\nabla\) on the vertical bundle V(Y) and an auxiliary linear connection \(\nabla_ 0\) on the base X are used, but the resulting form \(\Theta =\Theta (\nabla_ 0,Lv)\) is depending only on \(\nabla_ 0\). A discussion of the projectability of \(\Theta\) onto \(J^{2r-h}(p)\) \((h=1,...,r)\), interior variational formula for the density Lv, the concept of the Euler-Lagrange form and Hamiltonian extremals, and the common case of the Noether theorem appear as the resulting achievement of the article.
Mathematics Subject Classification ID: 49Q99
Mathematics Subject Classification ID: 58A15
Mathematics Subject Classification ID: 58A20
Mathematics Subject Classification ID: 53B05
Mathematics Subject Classification ID: 70H99
zbMATH DE Number: 3979896
zbMATH Keywords: fibred manifold
zbMATH Keywords: Poincaré-Cartan form
zbMATH Keywords: Lagrange density
zbMATH Keywords: linear connection
zbMATH Keywords: Euler-Lagrange form
zbMATH Keywords: Hamiltonian extremals
zbMATH Keywords: Noether theorem
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3979896