Commutator-free Cayley methods

TL;DR

This paper introduces a high-order, efficient, commutator-free Lie group integrator for non-autonomous differential equations on quadratic Lie groups, preserving physical invariants without costly matrix exponentials.

Contribution

It develops a novel integrator based on Cayley transforms that avoids nested commutators and matrix exponentials, improving computational efficiency and structure preservation.

Findings

The method is high-order and commutator-free.

It preserves Lie group structure and physical invariants.

It is computationally more efficient than existing methods.

Abstract

Differential equations posed on quadratic matrix Lie groups arise in the context of classical mechanics and quantum dynamical systems. Lie group numerical integrators preserve the constants of motions defining the Lie group. Thus, they respect important physical laws of the dynamical system, such as unitarity and energy conservation in the context of quantum dynamical systems, for instance. In this article we develop a high-order commutator free Lie group integrator for non-autonomous differential equations evolving on quadratic Lie groups. Instead of matrix exponentials, which are expensive to evaluate and need to be approximated by appropriate rational functions in order to preserve the Lie group structure, the proposed method is obtained as a composition of Cayley transforms which naturally respect the structure of quadratic Lie groups while being computationally efficient to evaluate. Unlike Cayley-Magnus methods the method is also free from nested matrix commutators.