Ulysses’ trip

Jan 26, 2020

Amazing Siteleaf

Article cover image with no practical usage I have been using Siteleaf to manage my blog. It is just convenient and amazing.

/update #update #jekyll

Jan 25, 2020

Hyperellipsoids in barycentric coordinates

Article cover image with no practical usage In this article, I introduce the barycentric coordinates: it is an elegant way to represent geometric shapes related to a simplex. By using it, given a simplex, we can construct a hyperellipsoid with the properties: its surface passes every vertex of the simplex, and its tangent hyperplane at each vertex is parallel to the hyperplane containing all other vertices.

/math #linear algebra #long paper

Jan 6, 2020

Use complex numbers as canonical variables

Article cover image with no practical usage In this article, I try exploring an idea: using complex numbers to combine pairs of canonical variables into complex variables: $\mathbf c\coloneqq\alpha\mathbf q+\mathrm i\beta\mathbf p$ . It turns out that we can write canonical equations $\frac{\mathrm d\mathbf c}{\mathrm dt}=-2\mathrm i\alpha\beta\frac{\partial\mathcal H}{\partial\mathbf c^*}$ , Poisson brackets $\left\{f,g\right\}=-2\mathrm i\alpha\beta \left(\frac{\partial f}{\partial\mathbf c}\cdot \frac{\partial g}{\partial\mathbf c^*}- \frac{\partial f}{\partial\mathbf c^*}\cdot \frac{\partial g}{\partial\mathbf c}\right)$ , and canonical transformations $\frac{\partial\mathbf c^*}{\partial\mathbf c'^*}= \frac{\partial\mathbf c'}{\partial\mathbf c}, \frac{\partial\mathbf c}{\partial\mathbf c'^*}= -\frac{\partial\mathbf c'}{\partial\mathbf c^*}$ in these complex numbers. Finally, I show two examples of using them in real problems: a free particle, and a harmonic oscillator.