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).