A local Lagrangian for MOND as modified inertia

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.