The following problem is of physical nature, but its core consists of pure mathematics, so I ask it here:
Suppose we have $n$ electric charges $q$ on a circle. They can move freely around it, but they cannot leave it. Then show that the system is in an equilibrium if and only if the charges are on the vertices of a regular polygon.
In an equilibrium, the force has to be radial. With angles $0=\vartheta_1<...<\vartheta_n<2\pi$ for the $n$ charges I worked out the condition: $$ \sum_{i=1,\ i\neq j}^{n}\frac{\sin(\vartheta_i-\vartheta_j)}{\sqrt{1-\cos(\vartheta_i-\vartheta_j)}}=0 $$ for all $1≤j≤n$. How can we deduce that $\vartheta_i=\frac{2\pi}{n}(i-1)$?
Hint: Without arguing by symmetry: the electric potential of a configuration of the charges is given by $$V(x_1,\ldots,x_n) = \sum_{i<j}\frac{1}{\|x_i-x_j\|}$$ where $x_1,\ldots,x_n$ are distinct points in $S^1\subset\mathbb{R}^2$. The system is in equilibrium if, and only if $V$ is at a critical point. Use Lagrange multipliers.