The Legendre-like operators on tuples of Lagrangians and functions

Abstract

Let \(Y\) be a fibred manifold with an \(m\)-dimensional basis \(M\).  We describe all Legendre-like operators \(C\), i.e. natural operators transforming tuples \((\lambda,g)\) of Lagrangians \(\lambda:J^sY\to\bigwedge ^mT^*M\) and functions \(g:M\to\mathbf{R}\) (resp. \(g:Y\to\mathbf{R}\)) into Legendre maps \(C(\lambda,g):J^{s}Y\to S^sTM\otimes V^*Y\otimes\bigwedge^m T^*M\). The most important example of such operators is the Legendre operator  (from the variational calculus) being  the one in question depending only on Lagrangians.