Existence of a variational principle for PDEs with symmetries and current conservation

Abstract

It is well-known from Noether's theorem that symmetries of a variational functional lead to conservation laws of the corresponding Euler-Lagrange equation. We reverse this statement and prove that a differential equation, which satisfies sufficiently many symmetries and corresponding conservation laws, leads to a variational functional, whose Euler-Lagrange equation is the given differential equation. Sufficiently many symmetries means that the set of symmetry vector fields satisfy span$\{V_p:V\in \text{Sym}\}=T_{p}E$, that is, they span the tangent space $T_{p}E$ at each point p of a fiber bundle E, which describes the dependent- and independent coordinates. Higher order coordinates are described by the jet bundle $J^{k}E$ and we require no span-assumptions on $T{J}^{k}E$. Our main theorem states that Noether's theorem can be reversed in this sense for second order differential equations, or more precisely, for so-called second order source forms on $J^{2}E$, which are required to write the differential equation as a weak formulation (every Euler-Lagrange equation is derived from a first variation, that is, from a weak formulation). Counter examples show that our theorem is sharp.