Closed-form modified Hamiltonians for integrable numerical integration schemes

TL;DR

This paper constructs explicit nonlinear integrable systems with closed-form modified Hamiltonians, enabling long-term accurate symplectic integration and highlighting the relevance of integrable mappings in geometric numerical methods.

Contribution

It provides explicit examples of integrable systems with convergent closed-form modified Hamiltonians, bridging integrable systems theory and geometric numerical integration.

Findings

Explicit nonlinear systems with closed-form modified Hamiltonians are constructed.

Modified Hamiltonians can be expressed in closed form for certain integrable mappings.

The study discusses the relevance of integrable mappings to geometric numerical integration.

Abstract

Modified Hamiltonians are used in the field of geometric numerical integration to show that symplectic schemes for Hamiltonian systems are accurate over long times. For nonlinear systems the series defining the modified Hamiltonian usually diverges. In contrast, this paper constructs and analyzes explicit examples of nonlinear systems where the modified Hamiltonian has a closed-form expression and hence converges. These systems arise from the theory of discrete integrable systems. We present cases of one- and two-degrees symplectic mappings arising as reductions of nonlinear integrable lattice equations, for which the modified Hamiltonians can be computed in closed form. These modified Hamiltonians are also given as power series in the time step by Yoshida's method based on the Baker-Campbell-Hausdorff series. Another example displays an implicit dependence on the time step which could be of relevance to certain implicit schemes in numerical analysis. In the light of these examples, the potential importance of integrable mappings to the field of geometric numerical integration is discussed.