A converse to Noether's theorem

The classical Noether's theorem states that symmetries of a variational functional lead to conservation laws of the corresponding Euler-Lagrange equation. It is a well-known statement to physicists with many applications. In the talk we investigate a reverse statement, namely 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. The aim of the talk is to provide some background of the so-called inverse problem of the calculus of variations and then to discuss some new results, for example, how to prove the reverse statement.