The Lagrangian Order-Reduction Theorem in Field Theories

Abstract

It is known that for every system of variational second order PDEs affine in second derivatives the Tonti Lagrangian is locally reducible to an equivalent first order Lagrangian (Order reduction Theorem). In this paper, a new proof is presented, based on investigation of closed forms related with variational equations, and an explicit formula for the first order Lagrangians arising by order reduction is found. The presented approach extends and completes the Order reduction Theorem by a geometric content and physical meaning of order reducibility: all variational second order PDEs affine in second derivatives admit a first-order covariant Hamiltonian formulation (Hamilton–De Donder equations), i.e. (under certain regularity conditions) carry a multisymplectic structure which is determined directly from the Euler–Lagrange expressions.