The conformal map w(z) transforms the trajectory with energy −B in potential U(z):=A∣dw/dz∣2 into the trajectory with energy −A in potential V(w):=B∣dz/dw∣2. I will prove this beautiful result and show some implications of it.
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=α/r that can repel the bullets. What regions are safe? The answer turns out to be the interior of a circular paraboloid.
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 u:=p2+p02p2−p02n^+p2+p022p0p, where u is the position of the particle on 3-sphere (a 4-dimensional vector),
p is the momentum of the original particle in Kepler problem, n^ is a vector perpendicular to the 3-dimensional hyperplane where p lies, and p0:=−2mE.
The conformal map transforms the trajectory with energy in potential into the trajectory with energy in potential . I will prove this beautiful result and show some implications of it.
The aliens intiated their attack to the earth! They shoot bullets with mass and speed from a far-awar planet. To defend, humans built a field that can repel the bullets. What regions are safe? The answer turns out to be the interior of a circular paraboloid.
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 is where is the position of the particle on 3-sphere (a 4-dimensional vector),
is the momentum of the original particle in Kepler problem, is a vector perpendicular to the 3-dimensional hyperplane where lies, and .