Characterization of approximate Partial Hamiltonian operators and related approximate first integrals

The approximate partial Hamiltonian systems for the characterization of associated approximate operators and first integrals are investigated. It is proved that if an approximate operator is an approximate partial Hamiltonian operator which provides a first integral, then its evolutionary representative is an approximate partial Hamiltonian operator providing a first integral. The extra operator conditions are provided for an approximate partial Hamiltonian operator in evolutionary form to yield an approximate first integral. Moreover, characterization of approximate partial Hamiltonian operators and associated first integral conditions are explicitly provided for the approximate Hamiltonian system. The theory is applied to a perturbed second-order ordinary differential equation (ODE), the perturbed orbit equation and perturbed weakly coupled nonlinear oscillators.