On the algebraic properties of difference approximations of Hamiltonian systems

In this paper, we examine difference approximations for dynamic systems characterized by polynomial Hamiltonians, specifically focusing on cases where these approximations establish birational correspondences between the initial and final states of the system. Difference approximations are commonly used numerical methods for simulating the evolution of complex systems, and when applied to Hamiltonian dynamics, they present unique algebraic properties due to the polynomial structure of the Hamiltonian. Our approach involves analyzing the conditions under which these approximations preserve key features of the Hamiltonian system, such as energy conservation and phase-space volume preservation. By investigating the algebraic structure of the birational mappings induced by these approximations, we aim to provide insights into the stability and accuracy of numerical simulations in identifying the true behavior of Hamiltonian systems. The results offer a framework for designing efficient and accurate numerical schemes that retain essential properties of polynomial Hamiltonian systems over time.