Archive of posts with tag “complex”
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.
For a plane lattice, there is only a finite number of different rotational symmetries that are compatible with the discrete translational symmetry. For example, the 5-fold rotational symmetry is not one of them. Why is that? It turns out that whether an -fold symmetry is compatible with translational symmetry is the same as whether .
When someone asks you why it is here instead of or the other way around, you can say that this is just a convention. My professor of quantum mechanics once asked the class similar a question, and I replied with this letter.
Joukowsky transformation of a circle centered at of radius is a curve resembling a heart.
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.