Adjoint symmetries for time-dependent second-order equations

The authors extend part of their previous work on autonomous second-order systems (Sarlet et al., 1987) to time-dependent differential equations. The main subject of the paper concerns the notion of adjoint symmetries: they are introduced as a particular type of 1-form, whose leading coefficients satisfy the adjoint equations of the equations determining symmetry vector fields. It is shown that all interesting properties of adjoint symmetries, known from the autonomous theory, have their counterparts in the present framework. Of particular interest is a result establishing that Lagrangian systems seem to be the only ones for which there is a natural duality between symmetries and adjoint symmetries. A number of examples illustrate how the construction of adjoint symmetries of a given system can be explored in a systematic way.