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I am looking to do a celestial simulation using MOND for the final project of my intermediate mechanics course. From the Wikipedia page, it seems that preserving Newton's Third Law requires deriving a modified Gauss's Law for Gravity from a Lagrangian. That equation is $$ \nabla \cdot(\mu(\frac{|\nabla \Phi|}{a_0})\nabla \Phi) = 4\pi G \rho $$ where $\mu(x) = \frac{dF(x^2)}{dx}$.

I don't really understand what $\mu(x)$ is. When modifying Newton's Law of Gravity, it is some kind of interpolating function, like $$\mu(\frac{a}{a_0}) = \frac{1}{\sqrt{1 + (\frac{a_0}{a})^2}}$$

Is that the same $\mu(x)$ used in the modified Poisson's Equation? If not, what exactly is that function and what does it being equal to $\frac{dF(x^2)}{dx}$ mean? Is $F$ itself another function? I want to find a gravitational field from this equation given some mass distribution, so I do need some specifics regarding what the actual modifying function is. Any help would be appreciated.

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    $\begingroup$ It's the same $\mu$. This explains $F$, but unfortunately doesn't express $\mu$ in terms of it, let alone motivate such an expression. $\endgroup$
    – J.G.
    Commented Sep 26, 2022 at 15:08
  • $\begingroup$ @J.G. So to calculate the MOND gravitational field, all I need is that $\mu$ I listed above and not $F$, right? If I understand the page you linked correctly, $F$ is required to derive the MOND Gauss's Law from the Lagrangian but is otherwise irrelevant to finding the field. $\endgroup$
    – Aryan Prasad
    Commented Sep 26, 2022 at 23:26
  • $\begingroup$ From the second link, under the "Nonrelativistic" heading, where $\phi$ is the standard Newtonian gravitational potential and F is a new dimensionless function. Does that answer your question? $\endgroup$
    – Kyle Kanos
    Commented Oct 1, 2022 at 13:36

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