A new approach to integrals of discretizations by polarization

TL;DR

This paper introduces a new algebraic method for deriving integrals of motion in polarization-based discretizations of polynomial vector fields, enhancing understanding of their conservation properties.

Contribution

It presents an innovative algebraic approach to identify integrals of motion in polarization discretizations, advancing the analysis of their conservation features.

Findings

Polarization discretizations preserve integrals of motion in Hamiltonian systems.

The new algebraic method simplifies derivation of conserved quantities.

Enhanced understanding of invariants in discretized polynomial ODEs.

Abstract

Recently, a family of unconventional integrators for ODEs with polynomial vector fields was proposed, based on the polarization of vector fields. The simplest instance is the by now famous Kahan discretization for quadratic vector fields. All these integrators seem to possess remarkable conservation properties. In particular, it has been proved that, when the underlying ODE is Hamiltonian, its polarization discretization possesses an integral of motion and an invariant volume form. In this note, we propose a new algebraic approach to derivation of the integrals of motion for polarization discretizations.