# Ulysses’ trip

## Posts

### Typesetting math in emails

After some comparison among solutions, I use KaTeX to typeset math in my emails.

- Categories: programming
- Tags: jekyll, ruby, tex

### Mapping from Kepler problem to free particle on 3-sphere

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 $\mathbf u:=\frac{p^2-p_0^2}{p^2+p_0^2}\hat{\mathbf n}+\frac{2p_0}{p^2+p_0^2}\mathbf p,$ where $\mathbf u$ is the position of the particle on 3-sphere (a 4-dimensional vector), $\mathbf p$ is the momentum of the original particle in Kepler problem, $\hat{\mathbf n}$ is a vector perpendicular to the 3-dimensional hyperplane where $\mathbf p$ lies, and $p_0:=\sqrt{-2mE}$.

- Categories: physics
- Tags: classical mechanics, canonical transformation, letter, kepler problem

### Solving linear homogeneous ODE with constant coefficients

By using power series, we can prove that the problem of solving linear homogeneous ODE with constant coefficients can be reduced to the problem of solving a polynomial with those coefficients. This article illustrates this point in detail, but it uses a very awful notation…

- Categories: math
- Tags: calculus, linear algebra, combinatorics, ode, long paper, from zhihu

### Thoughts on a middle school thermal physics problem

To heat an object with hot water, if we divide the water into more parts and use each part to heat the object one after another, the final temperature will be higher. If the number of parts tends to infinity, then the final temperature will tend to the limit $T+\mathrm e^{-\frac C{C_0}}\left(T_0-T\right)$, where $T$ is the initial temperature of the object, $T_0$ is the temperature of the hot water, $C$ is the heat capacity of the object, and $C_0$ is the heat capacity of the hot water.

- Categories: physics
- Tags: calculus, number sequence, from zhihu

### I created a community rhythm game called Dododo

I have created a community rhythm game called Dododo. It is a rhythm game with musical rhythm notations.

- Categories: update
- Tags: update, rhythm game

### New website for MOIS Project (club)

I have built a website for our club called MOIS Project. It is a club related to music and band.

### My activities in 2020 JDFZ Summer Session

JDFZ is hosting a summer session, where lectures covering a wide range of topics are held. Come and see what are my activities during this event.

### New website for Little Turings (club)

I, the president and one of the founders of Little Turings, created a website to distribute information about it. The website is bilingual (Chinese and English).

### Drawing a heart using Joukowsky transformation

Joukowsky transformation of a circle centered at $\left(1,1\right)$ of radius $1$ is a curve resembling a heart.

### Generalization of Euler–Lagrange equation

We may generalize Euler–Lagrange equation to higher dimensional optimization problems: find a function defined inside a region to extremize a functional defined as an integral over that region, with the constraint that the value of the function is fixed on the boundary of the region.

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