  Usage of a Hamiltonian perturbation theory for a nonconservative system is
counterintuitive and in general, a technical impossibility by definition.
However, the time-independent dual Hamiltonian formalism for the
nonconservative systems have opened the door for using various conservative
perturbation theories for investigating the dynamics of such systems. Here we
demonstrate that the Lie transform Hamiltonian perturbation theory can be
adapted to find the perturbative solutions and the frequency corrections for
the dissipative oscillatory systems. As a further application, we use the
perturbation theory to analytically calculate the Hannay angle for the van der
Pol oscillator's limit cycle trajectory when its parameters-the strength of the
nonlinearity and the frequency of the linear part-evolve cyclically and
adiabatically. For this van der Pol oscillator, we also numerically calculate
the corresponding geometric phase and establish its equivalence with the Hannay
angle.
