Poincaré-Cartan forms in higher order variational calculus on fibred manifolds
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.
scientific article; zbMATH DE number 3979896
scientific article