Abstract There is no general method which tells one how to eliminate fast variables in case of deterministic non-linear equations of motion. We consider the corresponding Liouville equation and search for a suitable projection operator. This approach has two appealing features: first, the problem formally reduces to the linear case and secondly the projection operator is fully determined by the structure of the equations and does not appear as an ad hoc choice. The formalism is applied to so-called rapid phase systems, systems for which a guiding center can be defined. The equations of motion for the guiding center are calculated to second order in the ratio of the two time scales.