Differential Operators, Symmetries and the Inverse Problem for Second-Order Differential Equations

With each second-order differential equation Z in the evolution space J1 (Mn+1) we associate, using the natural f(3,−1)-structure S and the f(3, 1)-structure K, a group G of automorphisms of the tangent bundle T(J1 (Mn+1))), with G isomorphic to a dihedral group of order 8. Using the elements of G and the Lie derivative, we introduce new differential operators on J1 (Mn+1) and new types of symmetries of Z. We analyze the relations between the operators and the “dynamical” connection induced by Z. Moreover, we analyze the relations between the various symmetries, also in connection with the inverse problem for Z. Both the approach based on the Poincarè–Cartan two forms and the one relying on the introduction of the so-called metrics compatible with Z are explicitly worked out.