  We propose a local Lagrangian for a point particle where its inertia part is
modified in the regime of small accelerations. For the standard gravitational
central force, it recovers the deep MOdified Newtonian Dynamics (MOND)
(accelerations $\ll a_0\approx 10^{-10}$m/s$^{2}$) equations of motion in the
case of a circular orbit. Perturbations to that turn on higher derivative
terms, leading to exponentially unstable solutions that must vanish in order to
account for the very small scattering of the Tully-Fisher relation. Unstable
solutions linearly growing with time remain valid for a characteristic
timescale of at least 3 billion years. We show that vertical perturbations
recover similar results to dark matter for old galaxies, but deviations could
be present for young ones. We also present ways to probe our approach and
describe some of its subtleties, such as the strong equivalence principle
(violated in general), the center of mass motion of a composite body, and how
in some cases it could overcome Ostrogradsky's instabilities (with naturally
occurring piecewise Lagrangians). Our main conclusions regarding our MOND-like
proposal are: (i) it constitutes a possible recipe where Ostrogradsky
instabilities could be "tamed"; (ii) it is a falsifiable approach in various
contexts and (iii) it might explain simultaneously some of the issues usual
modified gravity MOND and dark matter phenomenologies have difficulties
individually. These aspects seem relevant to start addressing practical ways to
differentiate modified gravity MOND from modified inertia and give insights
into alternative ways to tackle some astrophysical and cosmological puzzles.
