First Integrals of Dynamical Systems And Their Numerical Preservation

TL;DR

This paper explores the calculation of first integrals of a scalar differential equation using complex Lie symmetry methods and evaluates the effectiveness of symplectic numerical methods in preserving these integrals.

Contribution

It introduces a complex Lie symmetry approach to find first integrals and assesses their preservation through symplectic Runge-Kutta methods.

Findings

Structure-preserving methods yield qualitatively correct results

Good preservation of first integrals observed

Complex Lie symmetry method effectively computes integrals

Abstract

We calculate Noether like operators and first integrals of scalar equation y'' = -k^2 y using complex Lie symmetry method, by taking values of k and y to be real as well as complex. We numerically integrate the equations using a symplectic Runge-Kutta method and check for preservation of these first integrals. It is seen that these structure preserving numerical methods provide qualitatively correct numerical results and good preservation of first integrals is obtained.