On the variational calculus in fibered-fibered manifolds

W. M. Mikulski. "On the variational calculus in fibered-fibered manifolds." Annales Polonici Mathematici 89.1 (2006): 1-12. <http://eudml.org/doc/281037>.

@article{W2006,
abstract = {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.},
author = {W. M. Mikulski},
journal = {Annales Polonici Mathematici},
keywords = {fibered-fibered manifold; -jet prolongation bundle; th order Lagrangian; Euler morphism; natural operator},
language = {eng},
number = {1},
pages = {1-12},
title = {On the variational calculus in fibered-fibered manifolds},
url = {http://eudml.org/doc/281037},
volume = {89},
year = {2006},
}

TY - JOUR
AU - W. M. Mikulski
TI - On the variational calculus in fibered-fibered manifolds
JO - Annales Polonici Mathematici
PY - 2006
VL - 89
IS - 1
SP - 1
EP - 12
AB - 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.
LA - eng
KW - fibered-fibered manifold; -jet prolongation bundle; th order Lagrangian; Euler morphism; natural operator
UR - http://eudml.org/doc/281037
ER -