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• # Artykuł - szczegóły

## Annales Polonici Mathematici

2006 | 89 | 1 | 1-12

## On the variational calculus in fibered-fibered manifolds

EN

### Abstrakty

EN
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 $𝓔(λ):J^{r+s,2s,r+p}Y → 𝓥*Y ⊗ ⋀^{dimX}T*X$ 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 $𝓔(λ)∘ j^{r+s,2s,r+p}σ=0$. Next we study the naturality of the "Euler" morphism. We prove that any natural operator of the "Euler" morphism type is c𝓔(λ), c ∈ ℝ, provided dim X ≥ 2.

1-12

wydano
2006

### Twórcy

autor
• Institute of Mathematics, Jagiellonian University, Reymonta 4, 30-059 Kraków, Poland