If both masses are moving then why only calculate the gravitational potential energy of only one mass though both are acted by the conservative forces

Question

Suppose a two mass $M$ and $m$ are acted upon by the gravitational force, so both has conservative force that is doing a work, but we only take one conservative force and simply calculate the work done by one of the conservative forces, why? Why dont we calclate potential forces for each mass as both act upon by the conservative forces and add up the value.

Answer 1

The gravitational potential energy is associated with the gravitational field. If you were to count this energy twice, you would not observe conservation of energy as your system evolves.

Consider your masses $M>m$ initially at rest with some separation $r_1$, so total mechanical energy

$$ E_1=-\frac{GMm}{r_1}$$

They fall towards each other, and later have

$$ E_2=\frac{P^2}{2M}+\frac{p^2}{2m}-\frac{GMm}{r_2}$$

Conservation of momentum gives $\vec P=-\vec p$. In the limit $M\gg m$, such as when one mass is the Earth and the other is a piece of toast, you can see that “all” of the kinetic energy is acquired by the low-mass object, and that therefore including the potential energy twice would mispredict whether the toast lands buttered-side-down. As you approach $M\approx m$, there is no point where it suddenly becomes appropriate to double the potential energy.

Answer 2

Why dont we calclate potential forces for each mass

We do! If the gravitational potential of the two-mass system is $\Phi(r_1, r_2)$ then the gravitational forces on those masses are $$F_1=-\frac{\partial \Phi}{\partial r_1}$$ $$F_2=-\frac{\partial \Phi}{\partial r_2}$$

as both act upon by the conservative forces and add up the value.

Again, we do! If we want to calculate the total force that acts on the two-mass system, we add the individual forces together. But conservative as they are, the sum of all forces is zero, which is nothing but momentum conservation of the system.

Getting zero might seem less interesting than you thought, but actually null results are the most interesting in all science because zero can be measured to arbitrary relative precision. If anything in free space starts accelerating ever so small, you just need to wait a sufficiently long time to notice the effect. Thus, by waiting long enough next to a system at rest, and never seeing any movement, you can make sure that momentum conservation holds true to really fantastic accuracy.