Construction of adjoint symmetries for systems of second-order and mixed first- and second-order ordinary differential equations

We recall the notion of adjoint symmetries for second-order ordinary differential equations and sketch a recent evolution in the coordinate free description of this concept. We further indicate how the theory can be generalized to mechanical systems with nonholonomic constraints. For both cases, a theorem is presented which links a subclass of adjoint symmetries to first integrals (and Lagrangians). We then discuss how the theory can be used for a systematic construction of first integrals and how the resulting algorithm can be implemented in a computer algebra environment. The example of the last section can be used as a benchmark for testing the performance of programs for the automatic computation of symmetries.