# Coulomb's choreography

Exact solutions for many-body problems are rarely encountered in physics. The following problem deals with a non-trivial motion of four charges. Due to the symmetry of the problem it is possible to determine the trajectories of the charges analytically.

Four identical particles with mass $$m$$ and charge $$+q$$, orbit a charge $$-q$$ as shown in the figure. The four positive charges always form a square of side $$l(t)$$ while the negative charge stays at rest at the center of the square. The motion of the charges is periodic with period $$T$$. That is, if the vectors $$\vec{r}_k(t)$$, $$k=\{1,2,3,4\}$$, describe the position of the charges then we have that $\vec{r}_{k}(t+T)=\vec{r}_{k}(t).$ It is also known that the side of the square oscillates between $$l_{\textrm{min}}=\frac{1}{4}L_{0}$$ and $$l_{\textrm{max}}=L_{0}$$. Determine the period T in seconds if the parameters $$q, m$$ and $$L_{0}$$ satisfy the relation $\frac{k q^{2}}{m L_{0}^{3}}=10^{4}~\mbox{s}^{-2}, \quad \textrm{where} \quad k=\frac{1}{4\pi \epsilon_{0}}.$

Details and assumptions

• Ignore any radiation from the charges.
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