The Transition from Regular to Irregular Motions, Explained as Travel on Riemann Surfaces

Abstract

We introduce and discuss a simple Hamiltonian dynamical system, interpretable as a 3-body problem in the complex plane and providing the prototype of a mechanism explaining the transition from regular to irregular motions as travel on Riemann surfaces. The interest of this phenomenology -- illustrating the onset in a deterministic context of irregular motions -- is underlined by its generality, suggesting its eventual relevance to understand natural phenomena and experimental investigations. Here only some of our main findings are reported, without detailing their proofs: a more complete presentation will be published elsewhere.

26 pages, 2 figures

References in corpus (1)

• Periodic solutions of a many-rotator problem in the plane. II. Analysis of various motions

Cited by in corpus (13)

• Complex Correspondence Principle
• Newtonian dynamics in the plane corresponding to straight and cyclic motions on the hyperelliptic curve ${\mu }^2={\nu }^n-1,n\in \mathbb{Z}$: ergodicity, isochrony, periodicity and fractals
• Asymptotically isochronous systems
• Towards a Theory of Chaos Explained as Travel on Riemann Surfaces
• Quantum tunneling as a classical anomaly
• Probability Density in the Complex Plane
• Dynamical systems on infinitely sheeted Riemann surfaces
• Understanding complex dynamics by means of an associated Riemann surface
• Periodic orbits for classical particles having complex energy
• Abelian Chern-Simons Vortices and Holomorphic Burgers' Hierarchy
• Solvable and/or integrable many-body models on a circle
• Poisson Structures for Aristotelian Model of Three Body Motion
• Bi-Hamiltonian Structure of Gradient Systems in Three Dimensions and Geometry of Potential Surfaces