A Connection Between Geometry and Dynamical Systems

Abstract

This paper deals with a geometric and systematic approach to the integration of a nonlinear dynamical system: the anisotropic harmonic oscillator in a radial quartic potential. We study this system from a different angle: a) We show, using a Lax-type representation of the Hamilton’s equations of motion, that the system is linearized in the jacobian variety of a smooth genus 2 hyperelliptic curve. b) We find via Kowalewski-Painlevé analysis the principal balances of the hamiltonian vector field defined by the hamiltonian and we show that the system is algebraic complete integrable. c) We also describe an explicit embedding of the abelian variety which completes the generic invariant surface, into projective space. d) We give a direct proof that the abelian variety obtained in this paper is dual to Prym variety and can also be seen as a double unramified cover of the jacobian variety of an hyperelliptic curve of genus 2. e) We show that at some special values of the parameters λ₁ and λ₂, we can describe elliptic solutions which are associated with two-gap elliptic solitons of the Korteweg-de Vries equation.

Keywords:

• Lax representation
• Hamiltonian
• Integrable systems
• Curves
• Abelian varieties
• Jacobian varieties
• Prym varieties