On the variational calculus in fibered-fibered manifolds

W. M. Mikulski

Annales Polonici Mathematici, Tome 89 (2006), p. 1-12 / Harvested from The Polish Digital Mathematics Library

Résumé

In this paper we extend the variational calculus to fibered-fibered manifolds. Fibered-fibered manifolds are surjective fibered submersions π:Y → X between fibered manifolds. For natural numbers s ≥ r ≤ q with r ≥ 1 we define (r,s,q)th order Lagrangians on fibered-fibered manifolds π:Y → X as base-preserving morphisms $λ:J^{r,s,q}Y →⋀^{dimX}T*X$. Then similarly to the fibered manifold case we define critical fibered sections of Y. Setting p=max(q,s) we prove that there exists a canonical “Euler” morphism $(λ):J^{r+s,2s,r+p}Y → *Y ⊗ ⋀^{dimX}T*X$ of λ satisfying a decomposition property similar to the one in the fibered manifold case, and we deduce that critical fibered sections σ are exactly the solutions of the “Euler-Lagrange” equations $(λ)∘ j^{r+s,2s,r+p}σ=0$. Next we study the naturality of the “Euler” morphism. We prove that any natural operator of the “Euler” morphism type is c(λ), c ∈ ℝ, provided dim X ≥ 2.

Détail

Publié le : 2006-01-01

Zbl 1106.58003

EUDML-ID : urn:eudml:doc:281037