A New Class of Solvable Many-Body Problems

Abstract

A new class of solvable $N$-body problems is identified. They describe $N$ unit-mass point particles whose time-evolution, generally taking place in the complex plane, is characterized by Newtonian equations of motion "of goldfish type" (acceleration equal force, with specific velocity-dependent one-body and two-body forces) featuring several arbitrary coupling constants. The corresponding initial-value problems are solved by finding the eigenvalues of a time-dependent $N\times N$ matrix $U(t)$ explicitly defined in terms of the initial positions and velocities of the $N$ particles. Some of these models are asymptotically isochronous, i.e. in the remote future they become completely periodic with a period $T$ independent of the initial data (up to exponentially vanishing corrections). Alternative formulations of these models, obtained by changing the dependent variables from the $N$ zeros of a monic polynomial of degree $N$ to its $N$ coefficients, are also exhibited.