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BruceW

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I've been thinking about a certain problem for a while now. And that is a Lagrangian formulation of Newtonian gravity. I know there is a Lagrangian formulation for general relativity. But I was hoping to find a Lagrangian for Newtonian gravity instead (for some continuous mass distribution). Anyway, the closest thing I have been able to find is this equation (from wikipedia):

[tex]\mathcal{L} = -\rho \phi - \frac{1}{8\pi G} |\vec{g}|^2[/tex]

Where ##\rho## is mass density and ##\phi## is the gravitational potential, i.e.

[tex]\vec{g} = - \nabla \phi [/tex]

So, anyway, it says on wikipedia that you can vary this Lagrangian density with respect to changes in the gravitational field, to get Newton's law of gravity. This is true. But since we are only varying the gravitational field, and not the mass density, this means that we are keeping the mass density fixed! So this equation is not a true Lagrangian density for Newtonian gravity. In the true Lagrangian density, you would be able to vary both the mass density and the gravitational field, and this would give you both Newton's law of gravity, and some other equations (possibly conservation of momentum and energy?)

I was looking around for a while, but I haven't been able to find the true Lagrangian density for Newtonian gravity. I guess one way you could derive it, is to simply use the general relativity Lagrangian density, and then make some kind of low-speed, weak-gravity approximations, to get a Lagrangian density for Newtonian gravity. But that seems like quite a long way to go about it.

Thanks for reading, and hopefully some other members of PF have been thinking about this too. Or maybe know the answer?