Variational problems of some second order Lagrangians given by Pfaff forms

In this paper we study the dynamics of some second order Lagrangians that come from Pfaff forms, i.e. differential forms on tangent bundles. In the non-singular case, mainly considered in the paper, the generalized Euler-Lagrange equation is a third order differential equation. We prove that the solutions of the differential equations of motion of a charge in a field and the Euler equations of a rigid body can be obtained as particular solutions of suitable Pfaff forms, with non-negative second variations along their solutions. A non-standard Hamiltonian approach is also considered in the non-singular case, using energy functions associated with suitable semi-sprays. M.S.C. 2010: 70G45, 70H03, 70H06, 70H07, 70H30, 70H50, 53C80.