Isochronous dynamical systems

Abstract

An isochronous dynamical system is characterized by the existence of an open domain of initial data such that all motions evolving from it are completely periodic with a fixed period (independent of the initial data). Taking advantage of a recently introduced trick, a (quite large) class of such systems is identified, with equations of motion ‘of Newtonian type.’ Subcases are exhibited in which the equations of motion are Hamiltonian, and/or are interpretable as the equations of motion of a many-body problem, possibly with one- and two-body forces only, possibly invariant under rotations and/or translations, possibly constituting a one-parameter deformation of a classical physical problem (such as the many-body gravitational problem in ordinary three-dimensional space).

Keywords:

• Isochronicity
• Integrability
• Superintegrability
• Dynamical systems
• Hamiltonian systems
• Many-body problems
• 2000 Mathematics Subject Classifications: 37C27
• 37J35
• 37J45