Solvable and/or Integrable Many-Body Problems in The Plane, Obtained by Complexification

Abstract

The findings reported and outlined in Chap. 4 are all based on the idea to obtain two-dimensional models, i.e. models describing motions in the plane, from one-dimensional models, i.e. from models describing motions on the line, via a very simple trick: complexiflcation. How that works is explained in Sect. 4.1 The method is then illustrated by the discussion of a solvable model in Sect. 4.2 and its subsections, of some other solvable models in Sect. 4.3 and its subsections, and by a survey of solvable and/or integrable many-body problems in the plane obtainable by such an approach in Sect. 4.4 and its subsections. In Sect. 4.5 we investigate a many-rotator problem in the plane, which is rather closely related to the solvable model treated in Sect. 4.2.5, that only features completely periodic motions. A remarkable novelty is the possibility to treat variants of this solvable model which are instead, presumably, nonintegrable, yet exhibit sets of completely periodic motions which correspond to sets of initial data having nonvanishing measure in phase space. The mechanism which underlies this phenomenology, as analyzed in Sect. 4.5, brings to light an interesting connection among analyticity properties in the time variable, and integrable features of these motions, as manifested by their complete periodicity. Finally, Sect. 4.6 provides an outlook on future developments; the enterprising reader might like to browse through it immediately.