A CLASS OF INTEGRABLE HAMILTONIAN-SYSTEMS WHOSE SOLUTIONS ARE (PERHAPS) ALL COMPLETELY PERIODIC

We show that the dynamical system characterized by the (complex) equat ions of motion q(j)+i Omega q(j)=Sigma(k=1,k not equal j)(n)q(j)q(k)f( q(j)-q(k)), j=1,...,n, with f(x)=-lambda p'(lambda x)/[p(lambda x) - p (lambda mu)], is Hamiltonian and integrable, and we conjecture that al l its solutions q(j)(t), j=1,...,n are completely periodic, with a per iod that is a finite integral multiple of T=2 pi/Omega. Here n is an a rbitrary positive integer, Omega is an arbitrary (nonvanishing) real c onstant, p(y)=p(y\omega,omega') is the Weierstrass function (with arbi trary semiperiods omega,omega'), and lambda,mu are two arbitrary const ants; special cases are f(x)=2 lambda coth(lambda x)/[1 + r(2) sinh(2) (lambda x)], f(x)=2 lambda coth(lambda x>, f(x)=2 lambda/sinh(lambda x ), f(x)=2/[x(1 + lambda(2)x(2))], and of course f(x)=2/x. These findin gs, as well as the conjecture (which is shown to be true in some of th ese special cases), are based on the possibility to recast these equat ions of motion in the modified Lax form (L) over dot + i Omega L = [L, M] with L and M appropriate (n x n)-matrix functions of the n dynamica l variables q(j) and of their time-derivatives q(j). (C) 1997 American Institute of Physics.