Kato perturbation expansion in classical mechanics and an explicit expression for a Deprit generator

TL;DR

This paper connects Kato resolvent expansion to Deprit perturbation series in classical mechanics, providing an explicit formula for the Deprit generator and a systematic computational approach.

Contribution

It introduces an explicit expression for the Deprit generator using Kato series, systematizing perturbation series in classical mechanics.

Findings

Derived an explicit formula for the Deprit generator.

Established a connection between Kato series and perturbation operators.

Provided an effective computational algorithm based on Kato series.

Abstract

This work explores the structure of Poincare-Lindstedt perturbation series in Deprit operator formalism and establishes its connection to Kato resolvent expansion. A discussion of invariant definitions for averaging and integrating perturbation operators and their canonical identities reveals a regular pattern in a Deprit generator. The pattern was explained using Kato series and the relation of perturbation operators to Laurent coefficients for the resolvent of Liouville operator. This purely canonical approach systematizes the series and leads to the explicit expression for the Deprit generator in any perturbation order: \[G = - \hat{\mathsf S}_H H_i.\] Here, $\hat{\mathsf S}_H$ is the partial pseudo-inverse of the perturbed Liouville operator. Corresponding Kato series provides a reasonably effective computational algorithm. The canonical connection of perturbed and unperturbed averaging operators allows for a description of ambiguities in the generator and transformed Hamiltonian, while Gustavson integrals turn out to be insensitive to normalization style. Non-perturbative examples are used for illustration.