Kepler problem

Example

Let $k > 0$. The Kepler problem is the system $$r'' = -\frac{k\, r}{\norm{r}^3}$$ for a position $r(t) \in \R^3 \setminus \set{0}$, with velocity $v = r'$. It describes a body attracted to a fixed centre by an inverse-square force, such as a planet around the Sun. It is the Hamiltonian system with $H(r, v) = \frac{1}{2} \norm{v}^2 - \frac{k}{\norm{r}}$.

Three quantities are first integrals:

1. the energy $E = \frac{1}{2} \norm{v}^2 - k / \norm{r}$;
2. the angular momentum $L = r \times v$;
3. the Laplace–Runge–Lenz vector $A = v \times L - k\, r / \norm{r}$.

Check for $L$: $L' = v \times v + r \times r'' = 0 - \frac{k}{\norm{r}^3}\, r \times r = 0$.