On the variational calculus in fibered-fibered manifolds

W. M. Mikulski¹

¹Institute of Mathematics Jagiellonian University Reymonta 4 30-059 Krak/ow, Poland

Annales Polonici Mathematici, Tome 89 (2006) no. 1, pp. 1-12

Résumé

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 $\dim X\geq 2$.

Détail

DOI : 10.4064/ap89-1-1

Keywords: paper extend variational calculus fibered fibered manifolds fibered fibered manifolds surjective fibered submersions between fibered manifolds natural numbers geq leq geq define order lagrangians fibered fibered manifolds base preserving morphisms lambda y bigwedge dim *x similarly fibered manifold define critical fibered sections setting max prove there exists canonical euler morphism mathcal lambda y mathcal *y otimes bigwedge dim *x lambda satisfying decomposition property similar the fibered manifold deduce critical fibered sections sigma exactly solutions euler lagrange equations mathcal lambda circ sigma study naturality euler morphism prove natural operator euler morphism type mathcal lambda mathbb provided dim geq