Gravitation

Newtons Laws of Gravitation:

>figure 2 here 

$${\vec{F}}_g=-\frac{GmM}{r^2}\hat{r}$$

Gravitational force is a conservative force

$$\oint {\vec{F}}_g\cdot d\vec{r}=0$$ $$\oint {\vec{F}}_g\cdot d\vec{r}={\int }_S(\vec{\nabla }\times {\vec{F}}_g)d\vec{a}$$ $$\vec{\nabla }\times {\vec{F}}_g=0$$

The curl of gravitation is equal to 0 due to stokes theorem

$${\vec{F}}_g=-\vec{\nabla }u$$

Where $u$ is Gravitational potential energy given by:

$$u=GmM\left(-\frac{1}{r}\right)+\text{constant}$$

Decide that $u(r\to \infty )=0$

$$u(r)=-\frac{GmM}{r}$$

Gravitational Field $\vec{g}$

figure 3 here

$$\vec{g}=\frac{\vec{F}}{m}=-\frac{GM}{r^2}\hat{r}$$