Periodic Solutions of a Many-Rotator Problem in the Plane. II. Analysis of Various Motions

Abstract

International audienceVarious solutions are displayed and analyzed (both analytically and numerically) of a recently-introduced many-body problem in the plane which includes both integrable and nonintegrable cases (depending on the values of the coupling constants); in particular the origin of certain periodic behaviors is explained. The light thereby shone on the connection among integrability and analyticity in (complex) time, as well as on the emergence of a chaotic behavior (in the guise of a sensitive dependance on the initial data) not associated with any local exponential divergence of trajectories in phase space, might illuminate interesting phenomena of more general validity than for the particular model considered herein

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

“…This is the mechanism that accounts for the fact that, when the initial data are in certain sectors of their phase space (of course quite different from that characterized by the inequalities (12)), the resulting motion of the physical 3-body problem (1) is aperiodic, indeed nontrivially so: in such cases (as we show below) the aperiodicity is indeed associated with the coming into play of an infinite number of (square-root ) branch points of the corresponding solution of the auxiliary problem (9) and correspondingly with an infinite number of near misses experienced by the particles throughout their time evolution, this phenomenology being clearly characterized by a sensitive dependence on the initial data. This mechanism to explain the transition from regular to irregular motionsand in particular from an isochronous regime to one featuring unpredictable aspects -was already discussed [2] in the context of certain many-body models somewhat analogous to that studied herein. But those treatments were limited to providing a qualitative analysis such as that presented above and to ascertaining its congruence with numerical solutions of these models.…”

“…One may therefore conclude that our physical system (1) is isochronous with period 2 T, see (2). Indeed an isochronous system is characterized by the property to possess one or more open sectors of its phase space, each having of course full dimensionality, such that all motions in each of them are completely periodic with the same fixed period (the periods may be different in these different sectors of phase space, but must be fixed, i e. independent of the initial data, within each of these sectors): and in our case clearly (at least) all the motions characterized by initial data z n (0) such that min n,m=1,2,3; m =n…”

“…This 3-body problem is the prototype of a class of models [1][2][3] that feature a transition from very simple (even isochronous) to quite complicated motions characterized by a sensitive dependence both on the initial data and the parameters ("coupling constants") of the model. This transition can be explained as travel on Riemann surfaces.…”

New solvable many-body problems in the plane

“…13). Note that this model might as well be considered of "goldfish type", inasmuch as it features in the right-hand side of (1.22a) "velocity-dependent" forces -but the reduction to the standard "goldfish" model is, in this case, less obvious: it requires a more special choice of the (scalar ) source termsf…”

Two New Classes of Isochronous Hamiltonian Systems

“…Insertion of the third relation (3.15a), and of (3.17), in (3.12) implies that, in the limit (3.15b), 18) namely that the functions w n (τ ), hence as well the functions ζ n (τ ) (see (3.7)), are holomorphic in a circular disk centered at τ = 0, the radius of which can be made arbitrarily large by an appropriate assignment (see ( …”