On the Helmholtz operator of variational calculus in fibered-fibered manifolds

Volume 90 / 2007

W. M. Mikulski Annales Polonici Mathematici 90 (2007), 59-76 MSC: Primary 58A20. DOI: 10.4064/ap90-1-5

Abstract

A fibered-fibered manifold is a surjective fibered submersion $\pi :Y\to X$ between fibered manifolds. For natural numbers $s\geq r\leq q$ an $(r,s,q)$th order Lagrangian on a fibered-fibered manifold $\pi :Y\to X$ is a base-preserving morphism $\lambda :J^{r,s,q}Y\to \bigwedge ^{\mathop {\rm dim}X}T^*X$. For $p={\mathop {\rm max}}(q,s)$ there exists a canonical Euler morphism ${\mathcal E}(\lambda ):J^{r+s,2s,r+p}Y\to {\mathcal V}^*Y\otimes \bigwedge ^{\mathop {\rm dim}X}T^*X$ satisfying a decomposition property similar to the one in the fibered manifold case, and the critical fibered sections $\sigma $ of $Y$ are exactly the solutions of the Euler–Lagrange equation ${\mathcal E}(\lambda )\circ j^{r+s,2s,r+p}\sigma =0$. In the present paper, similarly to the fibered manifold case, for any morphism $B:J^{r,s,q}Y\to {\mathcal V}^*Y\otimes \bigwedge ^mT^*X$ over $Y$, $s\geq r\leq q$, we define canonically a Helmholtz morphism ${\mathcal H}(B) :J^{s+p,s+p,2p}Y\to {\mathcal V}^*J^{r,s,r}Y\otimes {\mathcal V}^*Y\otimes \bigwedge ^{\mathop {\rm dim} X}T^*X$, and prove that a morphism $B:J^{r+s,2s,r+p}Y\to {\mathcal V}^*Y\otimes \bigwedge T^*M$ over $Y$ is locally variational (i.e. locally of the form $B={\mathcal E}(\lambda )$ for some $(r,s,p)$th order Lagrangian $\lambda $) if and only if ${\mathcal H}(B)=0$, where $p={\mathop {\rm max}}(s,q)$. Next, we study naturality of the Helmholtz morphism ${\mathcal H}(B)$ on fibered-fibered manifolds $Y$ of dimension $(m_1,m_2,n_1,n_2)$. We prove that any natural operator of the Helmholtz morphism type is $c{\mathcal H}(B)$, $c\in {{\mathbb R}}$, if $n_2\geq 2$.

Authors

  • W. M. MikulskiInstitute of Mathematics
    Jagiellonian University
    Reymonta 4
    30-059 Kraków, Poland
    e-mail

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