The Euler-like operators on tuples of Lagrangians and functions on bases

Abstract

Let $\mathcal{FM}_{m,n}$ denote the category of fibered manifolds with $m$-dimensional bases and $n$-dimensional fibres and their fibered diffeomorphisms onto open images. We describe all $\mathcal{FM}_{m,n}$-natural  operators $C$ transforming tuples $(\lambda,g)$ of  Lagrangians $\lambda:J^sY\to\bigwedge ^mT^*M$ (or formal Lagrangians $\lambda:J^sY\to V^*J^sY\otimes\bigwedge ^mT^*M$) on $\mathcal{FM}_{m,n}$-objects $Y\to M$ and functions $g:M\to\mathbf{R}$ into Euler maps $C(\lambda,g):J^{2s}Y\to V^*Y\otimes\bigwedge^m T^*M$ on $Y$. The most important example of such $C$ is the Euler operator $E$ (from the variational calculus) (or the formal Euler operator $\mathbf{E}$) treated as the operator in question depending only on Lagrangians (or formal Lagrangians).