On the dynamic role of energy in underdamped particle motion

In his seminal work on stochastic particle transport in periodic potentials [1], claims that the overall energy, in underdamped conditions, plays the role of a slow dynamic variable, compared to position and velocity. The scope of this communication is to show that this claim is incorrect and that the conditions for the existence of a slow-invariant manifold fail in the low-friction limit. Nevertheless, the energy-based analysis developed by Risken to predict particle mobility remains substantially correct.