A new approach to the converse of Noether's theorem

Abstract
The concepts of vector fields and forms along a map are used to establish a condition characterising symmetries of the Hamiltonian system associated with a regular Lagrangian. This condition does not mention any second-order differential equation field but is expressed in terms of the geometry of the second-order tangent bundle. This result is also generalised to the case of Lagrangian functions depending on higher-order derivatives.