Another New Solvable Many-Body Model of Goldfish Type

[摘要] A new solvable many-body problem is identified. It is characterized by nonlinear Newtonian equations of motion (''acceleration equal force'') featuring one-body and two-body velocity-dependent forces ''of goldfish type'' which determine the motion of an arbitrary number $N$ of unit-mass point-particles in a plane. The $N$ (generally complex ) values $z_{n}( t) $ at time $t$ of the $N$ coordinates of these moving particles are given by the $N$ eigenvalues of a time-dependent $N\times N$ matrix $U( t) $ explicitly known in terms of the $2N$ initial data $z_{n}( 0) $ and $\dot{z}_{n}(0) $. This model comes in two different variants, one featuring 3 arbitrary coupling constants, the other only 2; for special values of these parameters all solutions are completely periodic with the same period independent of the initial data ('' isochrony ''); for other special values of these parameters this property holds up to corrections vanishing exponentially as $t\rightarrow \infty $ ('' asymptotic isochrony ''). Other isochronous variants of these models are also reported. Alternative formulations, obtained by changing the dependent variables from the $N$ zeros of a monic polynomial of degree $N$ to its $N$ coefficients, are also exhibited. Some mathematical findings implied by some of these results – such as Diophantine properties of the zeros of certain polynomials – are outlined, but their analysis is postponed to a separate paper.

[关键词] nonlinear discrete-time dynamical systems;integrable and solvable maps;isochronous discrete-time dynamical systems;discrete-time dynamical systems of goldfish type