我们提出了一个局部拉格朗日的点粒子，其惯性部分是在小加速度的政权修改。对于标准的引力中心力，它恢复了圆轨道情况下的深层修正牛顿动力学（MOND）（加速度$\ll a_0\approx 10^{-10}$ m/s $^{2}$）运动方程。对它的扰动会产生更高的导数项，导致指数不稳定的解，为了解释塔利-费舍尔关系的非常小的散射，这些解必须消失。随时间线性增长的不稳定解在至少30亿年的特征时间尺度内仍然有效。我们发现，垂直扰动恢复类似的结果，暗物质的老星系，但偏差可能存在于年轻的。我们还提出了一些方法来探索我们的方法，并描述了它的一些微妙之处，如强等效原理（违反一般），复合体的质心运动，以及在某些情况下，它如何克服奥斯特罗格拉茨基的不稳定性（自然发生的分段拉格朗日）。我们的主要结论，我们的MOND建议是：（i）它构成了一个可能的配方，奥斯特罗格拉茨基不稳定性可以“驯服”;（ii）这是一个可证伪的方法，在各种情况下，（iii）它可能同时解释一些问题通常修改重力MOND和暗物质现象学有困难的个别。这些方面似乎与开始解决区分修正的重力MOND和修正的惯性的实际方法有关，并提供了解决一些天体物理学和宇宙学难题的替代方法的见解。
英文摘要：
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.
