Perturbation to conformal invariance and adiabatic invariants of generalized perturbed Hamiltonian system with additional terms

Abstract

In this article, by considering the generalized perturbed Hamiltonian systems with additional terms, approximate conformal invariance and adiabatic invariants are studied. These invariants are obtained using approximate Lie point symmetries and without solving the structural equation. The necessary and sufficient condition for the relation between approximate conformal invariance and approximate Lie symmetry is obtained. The fundamental relationship between integrals and variables is presented herein, which is applicable to perturbed dynamic systems of both linear and non-linear nature. The approach is implemented on perturbed Hamiltonian systems of a generalized nature, featuring supplementary components of even dimensions. Additionally, the study derives the Hojman adiabatic invariants for generalized perturbed Hamiltonian systems with additional terms. Finally, an illustrative example is given to demonstrate the practical implementation of the findings.