Solution of the Goldfish N-Body Problem in the Plane with (Only) Nearest-Neighbor Coupling Constants All Equal to Minus One Half
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The (Hamiltonian, rotation- and translation-invariant) "goldfish" N-body problem in the plane is characterized by the Newtonian equations of motion ¨zn - i zn = 2 N m=1,m=n an,m zn zm (zn - zm) -1 , written here in their complex version, entailing the identification of the real "physical" plane with the complex plane. In this paper we exhibit in completely explicit form the solution of the initial-value problem for this N-body model in the special case in which the two-body interaction only acts among "nearest neighbors" (namely, only among particles whose labels differ by one unit: an,m = 0 unless |n - m| = 1) and the corresponding coupling constants all equal minus one half, an,n+1 = an+1,n = -1/2, n = 1, 2, ..., N - 1. This result implies that, if is a real nonvanishing constant, say, without loss of generality, > 0, then all the solutions of this N-body model are completely periodic indeed isochronous with period T = 2 / . An analogous conclusion holds as well for the model in which also the first and last particle interact with the same coupling constant, namely a1,N = aN,1 = -1/2 (rather than vanishing).
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Cite this article
TY - JOUR AU - Francesco Calogero PY - 2004 DA - 2004/02/01 TI - Solution of the Goldfish N-Body Problem in the Plane with (Only) Nearest-Neighbor Coupling Constants All Equal to Minus One Half JO - Journal of Nonlinear Mathematical Physics SP - 102 EP - 112 VL - 11 IS - 1 SN - 1776-0852 UR - https://doi.org/10.2991/jnmp.2004.11.1.7 DO - 10.2991/jnmp.2004.11.1.7 ID - Calogero2004 ER -