Isochronous systems are not rare

A survey will be given of isochronous systems, i. e. systems that oscillate with a fixed period (for largely arbitrary initial data). It will be shown how to manufacture many such models---mainly many-body problems whose time evolution is characterized by Newtonian equations of motion. In particular a technique will be described to modify fairly general models describing a time evolution, so that the modified systems are isochronous (with period T) yet mimic closely (or even exactly) the behavior of the unmodified system for a time interval Ttilde much smaller (or just smaller) than T. As a particularly remarkable example (joint work with F. Leyvraz), it will be shown how---given the (autonomous) Hamiltonian H describing the most general (standard) nonrelativistic many-body problem (arbitrary number N of particles, arbitrary masses, arbitrary dimensions of ambient space, forces depending arbitrarily from all the particle coordinates)---it is possible to construct another (also autonomous) Hamiltonian Htilde (in fact, an infinity of such Hamiltonians) featuring two additional arbitrary positive parameters T and Ttilde with T>Ttilde, and having the following two properties. (i) The new Hamiltonian Htilde yields, over the (arbitrarily long!) time interval Ttilde, a dynamical evolution very similar (or even identical) to that yielded by H. (ii) The Hamiltonian Htilde is isochronous: all its solutions (for arbitrary initial data) are completely periodic with period T. This finding raises (interesting?) questions about the difference among nonintegrable and integrable dynamic