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

Abstract

Abstract With each second-order differential equation Z in the evolution space J 1(M n+1) we associate, using the natural f(3, −1)-structure and the f(3, 1)-structure K, a group of automorphisms of the tangent bundle T (J 1(M n+1)), with isomorphic to a dihedral group of order 8. Using the elements of and the Lie derivative, we introduce new differential operators on J 1(M n+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 Poincare–Cartan two forms and the one relying on the introduction of the so-called metrics compatible with Z are explicitly worked out.