An integrable many-body problem

Source abstract

Some years ago, Mikhailov and Sokolov identified as integrable the neat system of two evolution equations \documentclass[12pt]{minimal}\begin{document}$\dot{U}=V^{2},$\end{document}U̇=V2, \documentclass[12pt]{minimal}\begin{document}$\dot{V}=U^{2}$\end{document}V̇=U2, where U ≡ U(t) and V ≡ V(t) are two N × N matrices, N is an arbitrarypositive integer, t (“time”) is the independent variable, and superimposed dots indicate the time derivatives. This entails, rather trivially, that the generic solution of the modified version of this model reading \documentclass[12pt]{minimal}\begin{document}$\dot{U}=V^{2}+i \omega U,$\end{document}U̇=V2+iωU, \documentclass[12pt]{minimal}\begin{document}$\dot{V}=U^{2}+i \omega V$\end{document}V̇=U2+iωV, with ω an arbitrary positive constant, is completely periodicwith period T = 2π/ω (or possibly a period which is an integer multiple of T): “isochrony.” Another, less trivial, consequence of their finding is the observation that the solution of the many-body problem characterized by the Hamiltonian system of N Newtonian evolution equations, \documentclass[12pt]{minimal}\begin{document}$\ddot{x}_{n}=-a^{2} x_{n}^{5}\break+\frac{g^{2}}{2}\sum _{m=1, m\ne n}^{N}[ ( x_{n}-x_{m}) ^{-3}+\left( x_{n}+x_{m}\right) ^{-3}], n=1,\ldots ,N$\end{document}ẍn=−a2xn5+g22∑m=1,m≠nN[(xn−xm)−3+xn+xm−3],n=1,...,N, where xn ≡ xn(t) are N scalar dependent variables and a, g are two arbitrary constants, is simply related to the evolution of the (appropriately rescaled) eigenvalues of a matrix simply related to an appropriate solution of the original Mikhailov-Sokolov integrablematrix evolution system, hence is itself integrable.