Generalization of approximate partial Noether approach in phase space

Abstract

The approximate partial Noether theorem proposed earlier for the ordinary differential equations (ODEs) (Naeem and Mahomed in Nonlinear Dyn 57(1–2):303–311, 2009) is generalized in phase space for the perturbed Hamiltonian-type systems. The notion of approximate partial Hamiltonian is developed. An approximate partial Hamiltonian gives rise to an approximate Hamiltonian-type perturbed dynamical system of first-order ODEs. An approximate Legendre-type transformation connects the approximate partial Lagrangian and what we term as approximate partial Hamiltonian. The formulas for approximate partial Hamiltonian operators determining equations and first integrals are provided explicitly. We have explained our approach with the help of simple illustrative example. Then, it is applied to establish the approximate first integrals, reductions and exact solutions of two perturbed cubically coupled Duffing–Van der Pol oscillators. Both resonant and nonresonant cases are considered.