Isotropic spherical mass stuck between two isotropic spherical greater masses

Question

This question is generated by the curiosity of mind while solving a numerical from a book of mechanics.
As the force of gravity is a vector quantity, it acts from center of mass which implies that it will act from center to center of an object.
I was thinking of a small object (isotropic, spherical, uniform) that gets stuck between two heavy spherical masses having the same geometry and property, and all three of them with no initial velocity in space.
Picture for demonstration attached.
My questions are "Where will the small mass move?
Is it toward Mass 1 or Mass 2? Or it will remain stuck forever?
Is Gravity continuous, that holds the object in equilibrium state?
Will the small object act as a medium to attract both masses that will make them revolve around the smaller one?

Mauro Giliberti · Accepted Answer

If you are working in Newtonian gravity (I assume you are, seeing the tag) and therefore you neglect all relativistic/quantum effects, then the central mass will stay in the same spot if placed exactly in the center of mass of the system.
The two bigger masses will be attracted towards the center by the small mass and the other big mass (and nothing stops them) while the small mass will be pulled in two opposite directions with the same magnitude, therefore (since the gravitational force, same as any force, is a vector) the two will cancel out.
For the conservation of angular momentum, if the system isn't rotating at the beginning it will continue to not rotate, and the movement will be completely linear along the axis passing through the center of the bodies.
As for the "is gravity continuous", it certainly is in Newtonian gravity, as well as in General Relativity (the best we can nowadays do to explain gravity). However, the question in a Quantum-Gravity environment is far from obvious: we just don't know.
Bonus: you can generalize this question with different, moving masses. It's the Lagrange points problem and the point you're asking about is L1 that can be found (numerically in general, analytically with some assumptions).

Agnius Vasiliauskas · Answer

Or it will remain stuck forever?

I believe NOT. Mass at the center can be approximated as point-like particle, subject under uncertainty principle :
$$Delta{p}~Delta{x} ge dfrac{hbar}{2}$$
In this situation we are pretty sure about central mass momentum, because side masses gravitational forces compensates each other : $mathbf F_1=-mathbf F_2$, so that central mass should not move. Thus momentum or kinetic energy must be close to zero :
$$frac {dvec p_{central}}{dt} = frac {dvec p_1}{dt} + frac {d vec p_2}{dt} = 0 $$
In such case $Delta{p} to 0$, but then where mass had that momentum is not clearly known : $Delta{x} gg Delta{p}$. And if so - over time central mass will accumulate critical distance change so that sooner or later will be attracted to one of big neighbors. Probably - the bigger neighboring masses,- sooner it will happen, because field fluctuations will be higher. "Stable point/position in time or space" is a dummy concept, there is no "ideally stable" positions in a universe. In addition to that, gravitational field strength should also fluctuate, due to the same uncertainty principle : $Delta{E_p}~Delta{t} ge dfrac{hbar}{2}$. Add another fact that gravitational waves (interaction force) travels at $c$ speed over bend space-time, due to gravity and you will see that there is no $100%$ stability guarantee in a universe.

Isotropic spherical mass stuck between two isotropic spherical greater masses

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