First integrals of ordinary difference equations which do not possess a variational formulation

TL;DR

This paper introduces a novel method for deriving first integrals of ordinary difference equations lacking variational formulations, demonstrated through solving a nonlinear third order ODE and its discretization.

Contribution

It presents a new approach to find first integrals for difference equations without Lagrangian or Hamiltonian structures, expanding analytical tools in this area.

Findings

Successfully derived three first integrals for a nonlinear third order ODE

Solved the invariant discretization of the ODE using the new method

Demonstrated the method's effectiveness on equations without variational formulations

Abstract

The paper presents a new method for finding first integrals of ordinary difference equations which do not possess Lagrangians, nor Hamiltonians. As an example we solve a third order nonlinear ordinary differential equation and its invariant discretization using three first integrals obtained using this method.