An Elementary Construction of Modified Hamiltonians and Measures of 2D Kahan Maps

Core Concepts

The author demonstrates an elementary construction method for the invariant of a cubic Hamiltonian system's KHK discretisation, showcasing its expressibility as the product of ratios of affine polynomials.

Abstract

The article explores the construction of invariants for KHK discretisations, focusing on factorisable cubic Hamiltonians. It presents a method to determine invariants based on hexagonal base points and their relationships. The study delves into specific examples like the Henon-Heiles potential and general non-factorisable cases, illustrating how invariants are derived from indeterminacy points. The analysis reveals insights into singular fibres configurations and their implications on integrability.

Stats

The invariant is represented by the ratio of two products of three affine polynomials: B1B2 B3B4 B5B6 / B2B3 B4B5 B6B1. The invariant preserves the triangle-like configuration observed in factorisable cubic Hamiltonians. In non-factorisable cases, such as the Henon-Heiles potential, additional lines are pushed to infinity as h approaches zero.

Quotes

"The structure is more special than the generic one, described in Corollary 2.12, and explains the additional triangle-like structure observed." "In particular we see that the lines in 40 are independent of h, so they are preserved by the continuum limit." "From a direct computation we see that the singular fibres configuration of the pencil associated to the invariant is of type A3 2⊕A1."

Key Insights Distilled From

An Elementary Construction of Modified Hamiltonians and Modified Measures of 2D Kahan Maps

by Giorgio Gubb... at arxiv.org 03-01-2024

https://arxiv.org/pdf/2309.00799.pdf

Deeper Inquiries

How do non-factorisable cases impact integrability compared to factorisable ones

Non-factorisable cases can have a significant impact on integrability compared to factorisable ones. In factorisable cases, where the Hamiltonian can be expressed as a product of linear factors, the system often exhibits special properties that make it easier to analyze and understand. These systems are more likely to possess additional invariants and conserved quantities, leading to integrability. On the other hand, non-factorisable cases present challenges as they do not easily lend themselves to analytical solutions or factorisations. This complexity can hinder efforts to identify invariants and conserved quantities, making it harder to determine the integrability of the system.

What implications do singular fibre configurations have on understanding discrete integrable systems

The configurations of singular fibres play a crucial role in understanding discrete integrable systems. Singular fibres represent points where certain geometric properties change abruptly within an elliptic curve or surface associated with the system's invariant. By analyzing these singular fibres, researchers can gain insights into the underlying structure and behavior of discrete integrable systems. The classification and study of singular fibre configurations provide valuable information about how different parts of the system interact and evolve over time. Understanding these configurations helps researchers identify key features that contribute to integrability and stability in discrete systems.

How can these findings be applied to higher-dimensional systems beyond two dimensions

The findings related to singular fibre configurations and their impact on discretization methods like Kahan-Hirota-Kimura (KHK) maps can be extended to higher-dimensional systems beyond two dimensions. By applying similar analytical techniques used in two-dimensional cases, researchers can explore how singular fibre patterns manifest in higher-dimensional spaces. Understanding how these configurations behave across multiple dimensions provides valuable insights into complex interactions within multidimensional discrete systems. This knowledge is essential for developing efficient algorithms for solving problems involving higher-dimensional dynamical systems while maintaining integrability properties.