TY - JOUR

T1 - New many-body problems in the plane with periodic solutions

AU - Gomez-Ullate, David

AU - Hone, Andy

AU - Sommacal, Matteo

PY - 2004

Y1 - 2004

N2 - In this paper we discuss a family of toy models for many-body interactions including velocity-dependent forces. By generalizing a construction due to Calogero, we obtain a class of N-body problems in the plane which have periodic orbits for a large class of initial conditions. The two- and three-body cases (N=2, 3) are exactly solvable, with all solutions being periodic, and we present their explicit solutions. For N≥4 Painlevé analysis indicates that the system should not be integrable, and some periodic and non-periodic trajectories are calculated numerically. The construction can be generalized to a broad class of systems, and the mechanism which describes the transition to orbits with higher periods, and eventually to aperiodic or even chaotic orbits, could be present in more realistic models with a mixed phase space. This scenario is different from the onset of chaos by a sequence of Hopf bifurcations.

AB - In this paper we discuss a family of toy models for many-body interactions including velocity-dependent forces. By generalizing a construction due to Calogero, we obtain a class of N-body problems in the plane which have periodic orbits for a large class of initial conditions. The two- and three-body cases (N=2, 3) are exactly solvable, with all solutions being periodic, and we present their explicit solutions. For N≥4 Painlevé analysis indicates that the system should not be integrable, and some periodic and non-periodic trajectories are calculated numerically. The construction can be generalized to a broad class of systems, and the mechanism which describes the transition to orbits with higher periods, and eventually to aperiodic or even chaotic orbits, could be present in more realistic models with a mixed phase space. This scenario is different from the onset of chaos by a sequence of Hopf bifurcations.

KW - mathematical physics

KW - statistical physics and nonlinear systems

UR - http://iopscience.iop.org/1367-2630/6/1/024/pdf/1367-2630_6_1_024.pdf

U2 - 10.1088/1367-2630/6/1/024

DO - 10.1088/1367-2630/6/1/024

M3 - Article

VL - 6

SP - 24

EP - 24

JO - New Journal of Physics

JF - New Journal of Physics

SN - 1367-2630

ER -