From the authors’ summary: The Poincaré-Cartan (PC) form of a Lagrangian on the bundle \(J^2=J^2(N,M)\) is, as a rule, defined on \(J^3\), thus leading to the non-equivalence between Euler-Lagrange and Hamilton-Cartan equations. This naturally leads to the problem of determining which Lagrangians have a PC form projectable onto \(J^2\), as they will then admit a second-order Hamiltonian formalism. There are specific examples of this phenomenon in field theory. This paper provides an explicit classification of such Lagrangians.