Energy of a triangle

For a triangle $$ABC$$, assume the points have potential $$\varphi_a,\varphi_b,\varphi_c$$. As shown in Fig. ???, $$AB'\perp AC, AC'\perp AB$$, combine the components of $$\nabla\varphi$$, we have

$$(\nabla\varphi)^2=\frac{1}{\sin^2 A}\left(\frac{\varphi_{ab}^2}{c^2}+\frac{\varphi_{ac}^2}{b^2}-\frac{2\varphi_{ab}\varphi_{ac}\cos A}{bc}\right)$$

The area of triangle is

$$S=\frac{bc\sin A}{2}$$

The energy of this triangle is

\begin{aligned} E_{\triangle}&\propto S(\nabla\varphi)^2\\ &\propto \frac{bc}{\sin A}\left(\frac{\varphi_{ab}^2}{c^2}+\frac{\varphi_{ac}^2}{b^2}-\frac{2\varphi_{ab}\varphi_{ac}\cos A}{bc}\right)\end{aligned}

$$\varphi_a$$求导得到：

$$\frac{\pp E_\triangle}{\pp \varphi_a}\propto \left(\frac{b}{c\sin A}-\cot A\right)\varphi_{ab}+\left(\frac{c}{b\sin A}-\cot A\right)\varphi_{ac}\label{a}$$

$$\frac{b}{c\sin A}-\cot A=\frac{b-c\cos A}{c\sin A}=\frac{a\cos C}{a\sin C}=\cot C$$

$$\frac{c}{b\sin A}-\cot A=\cot B$$

$$\frac{\pp E_\triangle}{\pp \varphi_a}\propto \varphi_{ab}\cot C+\varphi_{ac}\cot B \label{comp}$$

Stationary point equation for energy minimum

Let $$E$$ be the total energy and use $$i$$ to mark all triangles containing $$A$$. $$B_i, C_i$$ are other points in triangle $$i$$. In the stationary point, we have

$$\frac{\pp E}{\pp \varphi_a}=\sum_i\frac{\pp E_i}{\pp \varphi_a} \propto\sum_i \varphi_{ab_i}\cot C_i+\varphi_{ac_i}\cot B_i=0$$

So,

$$\sum_i\left(\cot C_i+\cot B_i\right)\varphi_{a}=\sum_i\left( \varphi_{b_i}\cot C_i+\varphi_{c_i}\cot B_i\right)$$
$$\Rightarrow\varphi_a=\frac{\sum_i \varphi_{b_i}\cot C_i+\varphi_{c_i}\cot B_i}{\sum_i\cot C_i+\cot B_i}$$

Assume we know coordinates of a triangle $$ABC$$. To calculate $$\cot A$$, we define $$\bm b=\overrightarrow{AB}, \bm c=\overrightarrow{AC}$$.

$$\cot A=\frac{bc\cos A}{bc\sin A}=\frac{\bm{b\cdot c}}{|\bm{b\times c}|}=\frac{\bm{b\cdot c}}{2S}$$

The denominator $$2S$$ is the same for a triangle.