On the exact inverse problem of the calculus of variations
  The article concerns the problem if a~given system of differential equations is identical with the Euler--Lagrange system of an~appropriate variational integral. Elementary approach is applied. The main results involve the determination of the first--order variational integrals related to the second--order Euler--Lagrange systems. 
Veronika Chrastinova, Vaclav Tryhuk
math.DG