In this paper we will discuss some features of the bi-Hamiltonian method for solving the Hamilton-Jacobi (H-J) equations by Separation of Variables, and make contact with the theory of Algebraic Complete Integrability and, specifically, with the Veselov-Novikov notion of algebro-geometric (AG) Poisson brackets. The bi-Hamiltonian method for separating the Hamilton-Jacobi equations is based on the notion of pencil of Poisson brackets and on the Gel'fand-Zakharevich (GZ) approach to integrable systems. We will herewith show how, quite naturally, GZ systems may give rise to AG Poisson brackets, together with specific recipes to solve the H-J equations. We will then show how this setting works by framing results by Veselov and Penskoi about the algebraic integrability of the Volterra lattice within the bi-Hamiltonian setting for Separation of Variables.
(2005). Gel'fand-Zakharevich systems and algebraic integrability: the Volterra lattice revisited [journal article - articolo]. In REGULAR & CHAOTIC DYNAMICS. Retrieved from http://hdl.handle.net/10446/20565
Gel'fand-Zakharevich systems and algebraic integrability: the Volterra lattice revisited
Pedroni, Marco
2005-01-01
Abstract
In this paper we will discuss some features of the bi-Hamiltonian method for solving the Hamilton-Jacobi (H-J) equations by Separation of Variables, and make contact with the theory of Algebraic Complete Integrability and, specifically, with the Veselov-Novikov notion of algebro-geometric (AG) Poisson brackets. The bi-Hamiltonian method for separating the Hamilton-Jacobi equations is based on the notion of pencil of Poisson brackets and on the Gel'fand-Zakharevich (GZ) approach to integrable systems. We will herewith show how, quite naturally, GZ systems may give rise to AG Poisson brackets, together with specific recipes to solve the H-J equations. We will then show how this setting works by framing results by Veselov and Penskoi about the algebraic integrability of the Volterra lattice within the bi-Hamiltonian setting for Separation of Variables.File | Dimensione del file | Formato | |
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