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.
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.
In this article, I try exploring an idea: using complex numbers to combine pairs of canonical variables into complex variables: c:=αq+iβp. It turns out that we can write canonical equations dtdc=−2iαβ∂c∗∂H, Poisson brackets
{f,g}=−2iαβ(∂c∂f⋅∂c∗∂g−∂c∗∂f⋅∂c∂g), and canonical transformations
∂c′∗∂c∗=∂c∂c′,∂c′∗∂c=−∂c∗∂c′ in these complex numbers. Finally, I show two examples of using them in real problems: a free particle, and a harmonic oscillator.
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.
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 .
In this article, I try exploring an idea: using complex numbers to combine pairs of canonical variables into complex variables: . It turns out that we can write canonical equations , Poisson brackets
, and canonical transformations
in these complex numbers. Finally, I show two examples of using them in real problems: a free particle, and a harmonic oscillator.