How are field theory Langrangians treated when some terms have 2 derivatives but others have only 1? Because the number of derivatives in a Lagrangian term is more easily even than odd, the discussions for newcomers to physics of breaking up Lagrangians into free theories and perturbations often does not give clear instruction on how to conceptualize and handle terms with a single derivative.
How should a mathematician speaking to physicists refer to the role played by terms with only one derivative (e.g. Chern Simons like terms) in the presence of a Yang Mills kinetic term? Would it be a kinetic term of lower order? A velocity dependent potential term? Would the interpretation change if the second order Yang - Mills like term were discarded? Is an m D m -like term with 1-derivative of a field m treated as part of the perturbation in perturbation theory or as generating a lower order summand of the 2nd order 'free' operator to be inverted?
Please feel free to rephrase the question if you understand what is being asked and the confusion is complicating the inquiry. Thanks in advance.