Polynomial Time Nondimensionalisation of Ordinary Differential Equations via their Lie Point Symmetries

TL;DR

This paper introduces a polynomial-time algorithm for nondimensionalising ordinary differential equations using Lie point symmetries, simplifying the equations by reducing parameters through invariant coordinates.

Contribution

It presents a novel polynomial-time algorithm for nondimensionalisation based on Lie group symmetries of ODEs, improving computational efficiency.

Findings

Algorithm's arithmetic complexity is polynomial in input size.

Method effectively reduces parameters in ODE systems.

Applicable to a broad class of differential equations.

Abstract

Lie group theory states that knowledge of a $m$-parameters solvable group of symmetries of a system of ordinary differential equations allows to reduce by $m$ the number of equation. We apply this principle by finding dilatations and translations that are Lie point symmetries of considered ordinary differential system. By rewriting original problem in an invariant coordinates set for these symmetries, one can reduce the involved number of parameters. This process is classically call nondimensionalisation in dimensional analysis. We present an algorithm based on this standpoint and show that its arithmetic complexity is polynomial in input's size.