THE SYMMETRY IN THE STRUCTURE OF DYNAMICAL AND ADJOINT SYMMETRIES OF 2ND-ORDER DIFFERENTIAL-EQUATIONS

With each second-order differential equation Z in the evolution space J(1)(M(n+1)) we associate, using a new dif1erential operator A(Z), four families of vector fields and 1-forms on J(1)(M(n+1)) providing a natural set-up for the study of symmetries, first integrals and the inverse problem for Z. We analyse the relations between the four families pointing out the symmetric structure of this set-up. When a Lagrangian for Z exists, the bijection between dynamical and dual symmetries is included in the whole context, suggesting the corresponding bijection between dual-adjoint and adjoint symmetries. As an application, we show how some results of the inverse problem can be framed naturally in this geometrical context.