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

Abstract

Let FMm,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 FMm,n-natural operators C transforming tuples (λ,g) of Lagrangians λ:JsY→⋀mT∗M (or formal Lagrangians λ:JsY→V∗JsY⊗⋀mT∗M) on FMm,n-objects Y→M and functions g:M→R into Euler maps C(λ,g):J2sY→V∗Y⊗⋀mT∗M on Y. The most important example of such C is the Euler operator E (from the variational calculus) (or the formal Euler operator E) treated as the operator in question depending only on Lagrangians (or formal Lagrangians).