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 r_k(t), k={1,2,3,4}, describe the position of the charges then we have that
r_k(t+T)=r_k(t).
It is also known that the side of the square oscillates between l_min=1/4L_0 and l_max=L_0. Determine the period T in seconds if the parameters q,m and L_0 satisfy the relation
(kq^2)/(m(L_0^3))=10^4 s^−2,where k=1/(4πϵ_0)
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