ABSTRACT The global gravitational potential, ϕ, is not commonly employed in the analysis of cosmological simulations, since its level sets do not show any clear correspondence to the underlying density field and its persistent structures. Here, we show that the potential becomes a locally meaningful quantity when considered from a boosted frame of reference, defined by subtracting a uniform gradient term $\phi _{\rm {boost}}(\boldsymbol{x}) = \phi (\boldsymbol{x}) + \boldsymbol{x} \cdot \boldsymbol{a}_0$ with acceleration $\boldsymbol{a}_0$. We study this ‘boosted potential’ in a variety of scenarios and propose several applications: (1) The boosted potential can be used to define a binding criterion that naturally incorporates the effect of tidal fields. This solves several problems of commonly used self-potential binding checks: (i) it defines a tidal boundary for each halo, (ii) it is much less likely to misidentify caustics as haloes (specially in the context of warm dark matter cosmologies), and (iii) performs better at identifying virialized regions of haloes – yielding to the expected value of 2 for the virial ratio. (2) This binding check can be generalized to filaments and other cosmic structures. (3) The boosted potential facilitates the understanding of the disruption of satellite subhaloes. We propose a picture where most mass loss is explained through a lowering of the escape energy through the tidal field. (4) We discuss the possibility of understanding the topology of the potential field in a way that is independent of constant offsets in the first derivative $\boldsymbol{a}_0$. We foresee that this novel perspective on the potential can help to develop more accurate models and improve our understanding of structure formation.