Matteo Sommacal

Periods of the Goldfish Many-Body Problem

Abstract Calogero’s goldfish N-body problem describes the motion of N point particles subject to mutual interaction with velocity-dependent forces under the action of a constant magnetic field transverse to the plane of motion. When all coupling constants are equal to one, the model has the property that for generic initial data, all motions of the system are periodic. In this paper we investigate which are the possible periods of the system for fixed N, and we show that there exist initial data that realize each of these possible periods. We then discuss the asymptotic behaviour of the maximal period for large particle number N.

The transition from regular to irregular motions, explained as travel on Riemann surfaces

We introduce and discuss a simple Hamiltonian dynamical system, interpretable as a three-body problem in the (complex) plane and providing the prototype of a mechanism explaining the transition from regular to irregular motions as travel on Riemann surfaces. The interest of this phenomenology—illustrating the onset in a deterministic context of irregular motions—is underlined by its generality, suggesting its eventual relevance to understand natural phenomena and experimental investigations. Here only some of our main findings are reported, without detailing their proofs: a more complete presentation will be published elsewhere.

Towards a theory of chaos explained as travel on Riemann surfaces

We investigate the dynamics defined by the following set of three coupled first-order ODEs: (z) over dot (n) + i omega z(n) = g(n+2)/z(n) - z(n+1) + g(n+1)/z(n) - z(n+2) It is shown that the system can be reduced to quadratures which can be expressed in terms of elementary functions. Despite the integrable character of the model, the general solution is a multiple-valued function of time (considered as a complex variable), and we investigate the position and nature of its branch points. In the semi-symmetric case (g(1) = g(2) not equal g(3)), for rational values of the coupling constants the system is isochronous and explicit formulae for the period of the solutions can be given. For irrational values, the motions are confined but feature aperiodic motion with sensitive dependence on initial conditions. The system shows a rich dynamical behaviour that can be understood in quantitative detail since a global description of the Riemann surface associated with the solutions can be achieved. The details of the description of the Riemann surface are postponed to a forthcoming publication. This toy model is meant to provide a paradigmatic first step towards understanding a certain novel kind of chaotic behaviour.

New many-body problems in the plane with periodic solutions

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 Painleve 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.

Solvable nonlinear evolution PDEs in multidimensional space involving trigonometric functions

A solvable nonlinear (system of) evolution PDEs in multidimensional space, involving trigonometric (or hyperbolic) functions, is identified. An isochronous version of this (system of) evolution PDEs in multidimensional space is also reported.

Solvable Nonlinear Evolution PDEs in Multidimensional Space

A class of solvable (systems of) nonlinear evolution PDEs in multidimensional space is discussed. We focus on a rotation-invariant system of PDEs of Schrodinger type and on a relativistically-invariant system of PDEs of Klein-Gordon type. Isochronous variants of these evolution PDEs are also considered.

Helical configurations of elastic rods in the presence of a long-range interaction potential

Recently, the integrability of the stationary Kirchhoff equations describing an elastic rod folded in the shape of a circular helix was proven. In this paper we explicitly work out the solutions to the stationary Kirchhoff equations in the presence of a long-range potential which describes the average constant force due to a Morse-type interaction acting among the points of the rod. The average constant force results to be parallel to the normal vector to the central line of the folded rod; this condition remarkably permits to preserve the integrability (indeed the solvability) of the corresponding Kirchhoff equations if the elastic rod features constant or periodic stiffnesses and vanishing intrinsic twist. Furthermore, we discuss the elastic energy density with respect to the radius and pitch of the helix, showing the existence of stationary points, namely stable and unstable configurations, for plausible choices of the featured parameters corresponding to a real bio-polymer.

Solvable nonlinear evolution PDEs in multidimensional space involving elliptic functions

A solvable nonlinear (system of) evolution PDEs in multidimensional space, involving elliptic functions, is identified, and certain of its solutions are exhibited. An isochronous version of this (system of) evolution PDEs in multidimensional space is also reported.