Algebraic discretization of time-independent Hamiltonian systems using a Lie-group/algebra approach

TL;DR

This paper introduces an algebraic Lie-group approach to discretize time-independent Hamiltonian systems, enabling exact solutions and error analysis without traditional time discretization, demonstrated on harmonic oscillator examples.

Contribution

It develops a novel algebraic formalism for discretizing Hamiltonian systems using Lie groups and algebras, avoiding time-derivative discretization and allowing exact solutions in integrable cases.

Findings

Exact discretization scheme for integrable systems

Error analysis method for numerical schemes

Application to harmonic oscillator examples

Abstract

In this paper, time-independent Hamiltonian systems are investigated via a Lie-group/algebra formalism. The (unknown) solution linked with the Hamiltonian is considered to be a Lie-group transformation of the initial data, where the group parameter acts as the time. The time-evolution generator (i.e. the Lie algebra associated to the group transformation) is constructed at an algebraic level, hence avoiding discretization of the time-derivatives for the discrete case. This formalism makes it possible to investigate the continuous and discrete versions of time for time-independent Hamiltonian systems and no additional information on the system is required (besides the Hamiltonian itself and the initial conditions of the solution). When the time-independent Hamiltonian system is integrable in the sense of Liouville, one can use the action-angle coordinates to straighten the time-evolution generator and construct an exact scheme (i.e. a scheme without errors). In addition, a method to analyse the errors of approximative/numerical schemes is provided. These considerations are applied to well-known examples associated with the one-dimensional harmonic oscillator.