Differential equation in Newton's law of gravitation I have a question in following differential equation:$$m(dv/dt)=-GMm/r^2$$
Using chain rule:$$dv/dt=(dv/dr)*(dr/dt)=v(dv/dr)$$
Then $$v(dv/dr)=-GMm/r^2$$
I don't understand that $dr/dt=v$ because there are two objects, I assume the other object's velocity is $v_2$.
After $\delta t$, distance between two objects' centres $r$ will reduce $\delta (v+v2)$
Then $$dr/dt = v+v_2$$
I don't know if it is wrong. Thanks to all people who offer helps :)
 A: This equation, like all equations arising in physics, is a model. It does not completely characterize the behavior of the physical system. Rather, it is a good enough approximation to be deemed as useful.
As you know, Newton's law of universal gravitation says that, between any two objects of mass $m_1,m_2$, separated by a distance $d$, there are equal and opposite gravitational forces of magnitude $\frac{Gm_1m_2}{d^2}$ between the two objects, pointing in the direction of their displacement vectors.
Let's now imagine a physical system in which we have these two objects in an isolated environment - empty space if you will.
It turns out that only one differential equation is simply not enough to describe this simple system.
To see why, let's make some constructions. Call the position occupied by object $1$ at time $t$ $\boldsymbol x_1$, and call the position occupied by object $2$ at time $t$ $\boldsymbol x_2$. The first object obeys the following differential equation:
$$\underset{\text{mass}}{m_1}~\underset{\text{accel.}}{\ddot{\boldsymbol x}_1}=\underbrace{Gm_1m_2~\frac{1}{\underset{\text{squared distance}}{\|\boldsymbol x_1-\boldsymbol x_2\|^2}}}_{\text{Net Force}}$$
Similarly, the the second object obeys a nearly identical equation:
$$m_2\ddot{\boldsymbol x}_2=Gm_1m_2~\frac{1}{\|\boldsymbol x_1-\boldsymbol x_2\|^2}$$
As you can see, the two equations are coupled - acceleration of the first object depends on the second, and vica versa. This system of coupled, nonlinear differential equations are difficult (impossible actually) to solve with pencil and paper, so what do we do? We make an assumption that simplifies the mathematics but remains suitably accurate to be useful.
When studying planetary motion, typically what we have is a large star of mass $M$ (say object 1) and a small planet of mass $m\ll M$ (say object 2). In this case, the acceleration of the small object is much much larger than that of the large object, and so, to simplify our model, we assume that the large object of mass $M$ is stationary. And, because we are assuming object 1 is motionless, we can, without loss of generality, assume it is at the origin. We also make the notation change $\boldsymbol x_2\to\boldsymbol x$ for clarity, and , with all these changes, our second equation simplifies to
$$\ddot{\boldsymbol x}=\frac{GM}{\|\boldsymbol x\|^2}$$
