Solution of certain integrable dynamical systems of Ruijsenaars-Schneider type with completely periodic trajectories (Q1586903)

This paper deals with the solution of the equations of motion \[ \ddot z_j+i\Omega\dot z_j=\sum^n_{k=1,k\neq j}\dot z_j\dot z_kf(z_j-z_k),\;j=1,\dots,n\tag{1} \] so-called Ruijsenaars-Schneider (RS) type. Very recently it was shown that (1) is solvable, if \[ (1)\;f(z)={2\over z},\quad (2)\;f(z)={2\over [z(1+ v^2 z^2)]},\quad (3)\;f(z)=2a \text{ctgh}(az), \] \[ (4)\;f(z)={2a\over\sinh(az)},\quad (5)\;f(z) =2 a \text{ctgh} (az)/[1+r^2 \sinh^2(az)], \] \[ (6)\;f(z)=-a{P'(az) \over\bigl[P (az)-P(ab) \bigr]}. \] It was conjectured, that (1) is solvable if and only if the constant \(\Omega\) is real and nonvanishing, and all trajectories \(z_j(t)\), \(j=1, \dots,n\) are periodic with period (at most) \(T'=Tn!\), \(T={2\pi\over\Omega}\) (2). The authors prove this conjecture for all cases listed above, except case (6). In contrast to previous results for this conjecture (case 1 and case 3) the approach of the authors allows them to treat problems under considerations more explicitly. The technique of solution rests on the ``Lax pair'' form.