Isochronous systems are not rare

by Francesco Calogero (University of Rome La Sapienza)

Wednesday 29 May 2013, 11:00 → 12:00 Europe/Stockholm

FA32

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 dynamics (all isochronous systems are integrable, indeed more than superintegrable), about the definition of chaotic behavior (including the apparent need to invent some such notion for a finite time interval), about the validity (say, for N≈10²³) of statistical mechanics and of the second principle of thermodynamics, about cosmology (say, for N≈10⁸⁵). It also demonstrates the impossibility to ascertain which dynamical theory is the correct one, out of an infinity of different theories predicting the same (exactly the same) evolution over an arbitrarily long time interval, but being qualitatively different (isochronous versus chaotic, integrable versus nonintegrable).
Time permitting, I will also report another recent finding obtained with F. Leyvraz: a macroscopic many-body system featuring undamped density oscillations.