KTH/Nordita/SU seminar in Theoretical Physics [before December 2013]

Wednesday 29 May 2013
from 11:00
to 12:00 at
FA32

Speaker :

Francesco Calogero (University of Rome La Sapienza)

Abstract :

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.