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.
We can derive the equation of motion for mechanical systems in a Galileo universe with ι time dimensions and χ space dimensions by generalizing the principle of relativity and Hamilton’s principle.
In this article, we will find that the inertia matrix naturally appears when we calculate the kinetic energy T or the angular momentum M of a rigid body. Then, we introduce the concept of principal inertia Jpri. We also study how the inertia matrix changes under translations and rotations and how those transformations may lead to conclusions that can help us simplify the calculation of inertia matrices.
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.
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.
We can derive the equation of motion for mechanical systems in a Galileo universe with time dimensions and space dimensions by generalizing the principle of relativity and Hamilton’s principle.
In this article, we will find that the inertia matrix naturally appears when we calculate the kinetic energy or the angular momentum of a rigid body. Then, we introduce the concept of principal inertia . We also study how the inertia matrix changes under translations and rotations and how those transformations may lead to conclusions that can help us simplify the calculation of inertia matrices.
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.