  The global gravitational potential, $\phi$, 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.
