First Integrals of Dynamical Systems And Their Numerical Preservation
For researchers in numerical integration, this is an incremental demonstration that symplectic methods preserve first integrals for a simple test equation.
The paper computes first integrals of a scalar ODE using complex Lie symmetry and tests their preservation via symplectic Runge-Kutta integration, finding good qualitative results.
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.