Abstract.
The first main result of this paper is the solution of the (complex) equations of motion \(\ddot{z}_j + {\rm i} \Omega \dot{z}_j = \sum^n_{k=1, k\not= j} \dot{z}_j\dot{z}_k f (z_j - z_k)$ with $f (z) = 2a~ {\rm cotgh}~ (az)/ [1 + r^2 {\rm sinh}^2 (az)]\), and the consequent confirmation of the conjecture that all the trajectories of this dynamical system are completely periodic with period (at most) \(T' = T n!, T = 2 \pi /\Omega\). We also discuss a symplectic reduction scheme which features new Lie-theoretic aspects for these systems. These developments are introduced here in the perspective of applying them in future studies to implement geometric quantization techniques.
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Submitted 09/06/98, revised 08/01/98, accepted 22/01/99
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Calogero, F., Françoise, J. Solution of Certain Integrable Dynamical Systems of Ruijsenaars-Schneider Type with CompletelyPeriodic Trajectories. Ann. Henri Poincaré 1, 173–191 (2000). https://doi.org/10.1007/PL00001000
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DOI: https://doi.org/10.1007/PL00001000