  We study the effective forces acting between colloidal particles trapped at a
fluid interface which itself is exposed to a pressure field. To this end we
apply what we call the ``force approach'', which relies solely on the condition
of mechanical equilibrium and turns to be in a certain sense less restrictive
than the more frequently used ``energy approach'', which is based on the
minimization of a free energy functional. The main goal is to elucidate the
advantages and disadvantages of the force approach as compared to the energy
approach. First, we derive a general stress-tensor formulation of the forces at
the interface and work out a useful analogy with 2D electrostatics in the
particular case of small deformations of the interface relative to its flat
configuration. We apply this analogy to compute the asymptotic decay of the
effective force between particles trapped at a fluid interface, extending the
validity of previous results. Second, we address the case of deformations of a
non-flat interface. We compute the deformation of a spherical droplet due to
the electric field of a charged particle trapped at its surface and conclude
that the interparticle capillary force is unlikely to explain certain recent
experimental observations. Finally we discuss the application to a generally
curved interface and show as an illustrative example that a nonspherical
particle deposited on an interface forming a minimal surface is pulled to
regions of larger curvature.

Soft Condensed Matter;Statistical Mechanics