Integrability of bi-Hamiltonian systems using Casimir functions and characteristic polynomials

TL;DR

This paper establishes that for a family of compatible Poisson brackets, the Casimir functions and characteristic polynomial coefficients commute, providing criteria for their completeness and generalizing previous algebraic constructions.

Contribution

It proves the commutativity of Casimir functions and polynomial coefficients in bi-Hamiltonian systems and offers a criterion for their completeness, extending prior algebraic frameworks.

Findings

Casimir functions commute with polynomial coefficients

Criteria for completeness of the family of functions

Generalization of previous algebraic constructions

Abstract

In this paper we prove that for a pencil of compatible Poisson brackets $\mathcal{P} = \left\{\mathcal{A} + \lambda\mathcal{B} \right\}$ the local Casimir functions of Poisson brackets $\mathcal{A} + \lambda \mathcal{B}$ and coefficients of the characteristic polynomial $p_{\mathcal{P}}$ commute w.r.t. all Poisson brackets of the pencil $\mathcal{P}$. We give a criterion when this family of functions is complete. These results generalize previous constructions of complete commutative subalgebras in the symmetric algebra $S(\mathfrak{g})$ of a finite-dimensional Lie algebra $\mathfrak{g}$ by A.S. Mishchenko & A.T. Fomenko, A.V. Bolsinov & P. Zhang and A.M. Izosimov.