Ulysses’ trip

Posts

  • Feb 29, 2020

    It is Feb 29 today!

    It is Feb 29 today. The date appears once for as long as 4 years!

  • Feb 9, 2020

    Monkey-patching graciously

    Monkey-patching is a powerful tool in programming. In this article, I used techniques of Ruby metaprogramming to define a series of methods def_after, def_before, etc. to help monkey-patching. They look graciously in that we can use it to shorten the codes for monkey-patching (avoiding aliasing and repeating codes).

  • Jan 26, 2020

    Amazing Siteleaf

    I have been using Siteleaf to manage my blog. It is just convenient and amazing.

  • Jan 25, 2020

    Hyperellipsoids in barycentric coordinates

    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.

  • Jan 6, 2020

    Use complex numbers as canonical variables

    In this article, I try exploring an idea: using complex numbers to combine pairs of canonical variables into complex variables: $\mathbf c:=\alpha\mathbf q+\mathrm i\beta\mathbf p$. 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.

  • Jan 1, 2020

    Giving birth to my own blog

    This is my first blog! I will share interesting things in my life here.

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