On the variational calculus in fibered-fibered manifolds

Volume 89 / 2006

W. M. Mikulski Annales Polonici Mathematici 89 (2006), 1-12 MSC: Primary 58A20. DOI: 10.4064/ap89-1-1


In this paper we extend the variational calculus to fibered-fibered manifolds. Fibered-fibered manifolds are surjective fibered submersions $\pi:Y\to X$ between fibered manifolds. For natural numbers $s\geq r\leq q$ with $r\geq 1$ we define $(r,s,q)$th order Lagrangians on fibered-fibered manifolds $\pi:Y\to X$ as base-preserving morphisms $\lambda:J^{r,s,q}Y\to\bigwedge^{{\rm dim}\, X}T^*X$. 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 $\mathcal E(\lambda):J^{r+s,2s,r+p}Y\to \mathcal V^*Y\otimes \bigwedge^{{\rm dim}\,X}T^*X$ of $\lambda$ satisfying a decomposition property similar to the one in the fibered manifold case, and we deduce that critical fibered sections $\sigma$ are exactly the solutions of the “Euler–Lagrange” equations ${\mathcal E}(\lambda)\circ j^{r+s,2s,r+p}\sigma=0$. Next we study the naturality of the “Euler” morphism. We prove that any natural operator of the “Euler” morphism type is $c\mathcal E(\lambda)$, $c\in\mathbb R$, provided $\dim X\geq 2$.


  • W. M. MikulskiInstitute of Mathematics
    Jagiellonian University
    Reymonta 4
    30-059 Krak/ow, Poland

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