Another New Solvable Many-Body Model of Goldfish Type
Abstract
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×N
matrix
U(t)
explicitly known in terms of the 2N initial data
z
n
(0)
and
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→∞
("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.