• ➕ Math

$$ \begin{aligned} \vec\nabla=\vec e_i\frac{\partial}{\partial r_i} \quad &; \quad \text{Div}=\vec\nabla \cdot \vec v=\text{scalar} \\ \text{Grad}=\vec\nabla U=\text{vector} \quad &;\quad \text{Curl}=\vec\nabla \times\vec v=\text{vector} \end{aligned} $$

• 🪠 Types of forces

Conservative force fields can be generated by a potential

Condition for conservative field $\vec\nabla \times \vec F=0$

Central force fields $\vec F=f(r)\,\hat r$

• 🌏 Gravity

$$ \vec \nabla g(r)=\frac{\partial g}{\partial r}\,\hat r $$

Gravitational field strength $\vec g(\vec r)=\frac 1m \vec F$ and gravitational potential $\Phi=\frac 1m U$

Gravitational potential

$$ \text d \Phi(\vec r)=-G_N \frac{\text dM(r')}{|\vec r-\vec r'|} $$

• 🚓 Motion for central potentials

  $$ E=\frac 12 \mu \dot r^2+U_\text{eff}(r)=\frac 12 \mu \dot r^2+ U(r)+\frac{L^2}{2\mu r^2} $$

  General solution for $U=-\alpha/r$ is

  $$ r=\frac{r_0}{1+\epsilon\cos(\theta)} $$

  where $r_0$ and $\epsilon$ are related to $E$ and $L$ and the solutions leads to the following

  Circle	Ellipse	Parabola	Hyperbola
  $\epsilon=0$	$0<\epsilon<1$	$\epsilon=1$	$\epsilon >1$

• ⛰️ Stationary points

In general cases sketch the effective potential $U_\text{eff}(r)$ in region $r>0$ by identifying and classifying the stationary points, crossing points and asymptotes

Stationary points are stable if $\frac{\text d^2 U_\text {eff}}{\text d r^2}>0$ at the stationary point and unstable if it is $<0$