## Archive of posts with tag “kepler problem”

### Defend our earth against aliens’ bullets!

The aliens intiated their attack to the earth! They shoot bullets with mass $m$ and speed $v$ from a far-awar planet. To defend, humans built a field $U=\alpha/r$ that can repel the bullets. What regions are safe? The answer turns out to be the interior of a circular paraboloid.

- Categories: physics
- Tags: calculus, classical mechanics, kepler problem, from zhihu

### Mapping from Kepler problem to free particle on 3-sphere

There is a canonical transform of the Kepler problem which is the same as the problem of motion of a free particle on 3-sphere. The explicit formula of the transform as well as some links about this topic is written in the article. The explicit formula for $E<0$ is $\mathbf u:=\frac{p^2-p_0^2}{p^2+p_0^2}\hat{\mathbf n}+\frac{2p_0}{p^2+p_0^2}\mathbf p,$ where $\mathbf u$ is the position of the particle on 3-sphere (a 4-dimensional vector), $\mathbf p$ is the momentum of the original particle in Kepler problem, $\hat{\mathbf n}$ is a vector perpendicular to the 3-dimensional hyperplane where $\mathbf p$ lies, and $p_0:=\sqrt{-2mE}$.

- Categories: physics
- Tags: classical mechanics, canonical transformation, letter, kepler problem