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The underdamped, non-linear, generalized Langevin equation is widely used tomodel coarse-grained dynamics of soft and biological materials. By means of aprojection operator formalism, we show under which approximations this equationcan be obtained from the Hamiltonian dynamics of the underlying microscopicsystem and in which cases it makes sense to introduce a potential of meanforce. We discuss shortcomings of previous derivations presented in theliterature and demonstrate the implications of our derivation for the structureof memory terms and their connection to generalized fluctuation-dissipationrelations. We show, in particular, that the widely used, simple structure whichcontains a potential of mean force, a memory term which is linear in theobservable, and a fluctuating force which is related to the memory term by afluctuation-dissipation relation, is neither exact nor can it, in general, bederived as a controlled approximation to the exact dynamics.

The Interplay between Memory and Potentials of Mean Force: A Discussion on the Structure of Equations of Motion for Coarse Grained Observables